1236:
720:
1231:{\displaystyle {\begin{aligned}\log \left(\sum _{n=1}^{\infty }{\frac {1}{n}}\right)&{}=\log \left(\prod _{p}{\frac {1}{1-p^{-1}}}\right)=-\sum _{p}\log \left(1-{\frac {1}{p}}\right)\\&=\sum _{p}\left({\frac {1}{p}}+{\frac {1}{2p^{2}}}+{\frac {1}{3p^{3}}}+\cdots \right)\\&=\sum _{p}{\frac {1}{p}}+{\frac {1}{2}}\sum _{p}{\frac {1}{p^{2}}}+{\frac {1}{3}}\sum _{p}{\frac {1}{p^{3}}}+{\frac {1}{4}}\sum _{p}{\frac {1}{p^{4}}}+\cdots \\&=A+{\frac {1}{2}}B+{\frac {1}{3}}C+{\frac {1}{4}}D+\cdots \\&=A+K\end{aligned}}}
3979:
74:
4278:
6419:
3271:
6728:
3688:
3987:
4965:
2997:
5917:
3644:
4728:
5616:
3974:{\displaystyle {\begin{aligned}\sum _{k=1}^{n}{\frac {1}{k^{2}}}&<1+\sum _{k=2}^{n}\underbrace {\left({\frac {1}{k-{\frac {1}{2}}}}-{\frac {1}{k+{\frac {1}{2}}}}\right)} _{=\,{\frac {1}{k^{2}-{\frac {1}{4}}}}\,>\,{\frac {1}{k^{2}}}}\\&=1+{\frac {2}{3}}-{\frac {1}{n+{\frac {1}{2}}}}<{\frac {5}{3}}\end{aligned}}}
4273:{\displaystyle {\begin{aligned}\log(n+1)&<\sum _{i=1}^{n}{\frac {1}{i}}\\&\leq \prod _{p\leq n}\left(1+{\frac {1}{p}}\right)\sum _{k=1}^{n}{\frac {1}{k^{2}}}\\&<{\frac {5}{3}}\prod _{p\leq n}\exp \left({\frac {1}{p}}\right)\\&={\frac {5}{3}}\exp \left(\sum _{p\leq n}{\frac {1}{p}}\right)\end{aligned}}}
240:
5764:
3266:{\displaystyle {\begin{aligned}\left(1+{\frac {1}{p_{1}}}\right)\left(1+{\frac {1}{p_{2}}}\right)\ldots \left(1+{\frac {1}{p_{s}}}\right)&=\left({\frac {1}{p_{1}}}\right)\left({\frac {1}{p_{2}}}\right)\cdots \left({\frac {1}{p_{s}}}\right)+\ldots \\&={\frac {1}{p_{1}p_{2}\cdots p_{s}}}+\ldots .\end{aligned}}}
2894:
3422:
676:
5391:
382:
2631:
100:
1528:
4960:{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }{\frac {1}{p_{n}}}&\geq \sum _{n=6}^{\infty }{\frac {1}{p_{n}}}\\&\geq \sum _{n=6}^{\infty }{\frac {1}{n\log n+n\log \log n}}\\&\geq \sum _{n=6}^{\infty }{\frac {1}{2n\log n}}=\infty \end{aligned}}}
514:
2730:
2334:
527:
4612:
2992:
1414:
5912:{\displaystyle {\frac {\text{odd}}{\text{even}}}+{\frac {1}{p_{n+1}}}={\frac {{\text{odd}}\cdot p_{n+1}+{\text{even}}}{{\text{even}}\cdot p_{n+1}}}={\frac {{\text{odd}}+{\text{even}}}{\text{even}}}={\frac {\text{odd}}{\text{even}}}}
1686:
4388:
3639:{\displaystyle {\begin{aligned}\log(n+1)&=\int _{1}^{n+1}{\frac {dx}{x}}\\&=\sum _{i=1}^{n}\underbrace {\int _{i}^{i+1}{\frac {dx}{x}}} _{{}\,<\,{\frac {1}{i}}}\\&<\sum _{i=1}^{n}{\frac {1}{i}}\end{aligned}}}
274:
2133:
2478:
4473:
725:
4721:
1310:
5067:
1421:
414:
5611:{\displaystyle \sum _{i=0}^{\infty }(x_{k})^{i}>\sum _{j=1}^{\infty }{\frac {1}{1+j(p_{1}p_{2}\cdots p_{k})}}>{\frac {1}{1+p_{1}p_{2}\cdots p_{k}}}\sum _{j=1}^{\infty }{\frac {1}{j}}=\infty }
2721:
1945:
4733:
3992:
3693:
3427:
3002:
2396:
2193:
77:
The sum of the reciprocal of the primes increasing without bound. The x axis is in log scale, showing that the divergence is very slow. The red function is a lower bound that also diverges.
5280:
5384:
5219:
3335:
4524:
235:{\displaystyle \sum _{p{\text{ prime}}}{\frac {1}{p}}={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+{\frac {1}{13}}+{\frac {1}{17}}+\cdots =\infty }
2902:
1326:
5972:, which is the product of all these primes. Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime
704:
Euler considered the above product formula and proceeded to make a sequence of audacious leaps of logic. First, he took the natural logarithm of each side, then he used the
5319:
691:. Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.
1607:
5655:
5135:
2889:{\displaystyle \sum _{i=1}^{n}{\frac {1}{i}}\leq \left(\prod _{p\leq n}\left(1+{\frac {1}{p}}\right)\right)\cdot \left(\sum _{k=1}^{n}{\frac {1}{k^{2}}}\right)=A\cdot B.}
4301:
2035:
5095:
3374:
6600:
4403:
6253:
3406:
5675:
5155:
6693:
4646:
6534:
1252:
4995:
671:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=\prod _{p}\left(1+{\frac {1}{p}}+{\frac {1}{p^{2}}}+\cdots \right)=\prod _{p}{\frac {1}{1-p^{-1}}}}
2445:
Here is another proof that actually gives a lower estimate for the partial sums; in particular, it shows that these sums grow at least as fast as
6544:
6757:
2648:
1883:
377:{\displaystyle \sum _{\scriptstyle p{\text{ prime}} \atop \scriptstyle p\leq n}{\frac {1}{p}}\geq \log \log(n+1)-\log {\frac {\pi ^{2}}{6}}}
2645:
is odd). Factor out one copy of all the primes whose β is 1, leaving a product of primes to even powers, itself a square. Relabeling:
6539:
2626:{\displaystyle i=q_{1}^{2{\alpha }_{1}+{\beta }_{1}}\cdot q_{2}^{2{\alpha }_{2}+{\beta }_{2}}\cdots q_{r}^{2{\alpha }_{r}+{\beta }_{r}},}
2341:
6194:
6118:
6049:
6708:
6299:
6246:
6688:
6590:
6580:
6206:
5957:
form, this partial sum cannot be an integer (because 2 divides the denominator but not the numerator), and the induction continues.
2472:
1850:
688:
261:
6698:
6182:
4628:
1547:
approaches infinity. It turns out this is indeed the case, and a more precise version of this fact was rigorously proved by
6703:
6605:
6239:
5621:
4968:
409:
4624:
397:
6752:
6731:
1523:{\displaystyle A={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+\cdots =\log \log \infty .}
6713:
6002:
6585:
6575:
6565:
6595:
5224:
509:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots =\infty }
6025:(1737). "Variae observationes circa series infinitas" [Various observations concerning infinite series].
5991:
5324:
5160:
17:
3276:
5969:
6680:
6502:
2329:{\displaystyle x-|M_{x}|\leq \sum _{i=k+1}^{\infty }|N_{i,x}|<\sum _{i=k+1}^{\infty }{\frac {x}{p_{i}}}}
6342:
6289:
4640:
1560:
84:
6549:
6294:
6078:
519:
6073:
5677:, and since the tails of a convergent series must themselves converge to zero, this proves divergence.
2471:
can be uniquely expressed as the product of a square-free integer and a square as a consequence of the
6660:
6497:
6266:
3661:
1582:
6640:
6507:
5985:
5689:
of the reciprocals of the primes eventually exceed any integer value, they never equal an integer.
5285:
1826:
253:
2723:
where the first factor, a product of primes to the first power, is square free. Inverting all the
6481:
6466:
6438:
6418:
6357:
6160:
5627:
5100:
6570:
6103:, Vol. 78, No. 3 (Mar. 1971), pp. 272-273. The half-page proof is expanded by William Dunham in
6198:
73:
6670:
6471:
6443:
6397:
6387:
6367:
6352:
6202:
5996:
4980:
3417:
1586:
1532:
It is almost certain that Euler meant that the sum of the reciprocals of the primes less than
389:
43:
5074:
3344:
6655:
6476:
6402:
6392:
6372:
6274:
6152:
4607:{\displaystyle \lim _{n\to \infty }\left(\sum _{p\leq n}{\frac {1}{p}}-\log \log n\right)=M}
91:
2987:{\displaystyle {\frac {1}{i}}={\frac {1}{p_{1}p_{2}\cdots p_{s}}}\cdot {\frac {1}{b^{2}}},}
1409:{\displaystyle \log \left({\frac {1}{1-x}}\right)=\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}.}
6433:
6362:
5282:
contains at least one term for each reciprocal of a positive integer not divisible by any
3682:
1593:
3383:
6665:
6650:
6645:
6324:
6309:
6187:
6044:
6022:
5660:
5140:
1971:
245:
408:
First, we describe how Euler originally discovered the result. He was considering the
6746:
6630:
6304:
6164:
6069:
4478:
1681:{\displaystyle \sum _{i=k+1}^{\infty }{\frac {1}{p_{i}}}<{\frac {1}{2}}\qquad (1)}
1564:
1548:
705:
684:
257:
6635:
6377:
6319:
6224:
6143:
Lord, Nick (2015). "Quick proofs that certain sums of fractions are not integers".
4383:{\displaystyle \log \log(n+1)-\log {\frac {5}{3}}<\sum _{p\leq n}{\frac {1}{p}}}
88:
34:
This article uses technical mathematical notation for logarithms. All instances of
5137:
contains at least one term for each reciprocal of a positive integer with exactly
2128:{\displaystyle \{1,2,\ldots ,x\}\setminus M_{x}=\bigcup _{i=k+1}^{\infty }N_{i,x}}
396:) indicates that the divergence might be very slow, which is indeed the case. See
6382:
6329:
5686:
1766:
268:
5976:
divide the denominator. Thus the expression is irreducible and is non-integer.
4468:{\displaystyle \sum _{k=1}^{\infty }{\frac {1}{k^{2}}}={\frac {\pi ^{2}}{6}}}
2452:. The proof is due to Ivan Niven, adapted from the product expansion idea of
6314:
1714:
262:
divergence of the sum of the reciprocals of the integers (harmonic series)
6262:
2460:
always represents a sum or product taken over a specified set of primes.
6156:
6231:
4716:{\displaystyle p_{n}<n\log n+n\log \log n\quad {\mbox{for }}n\geq 6}
1596:
4393:
1551:
in 1874. Thus Euler obtained a correct result by questionable means.
249:
2453:
1314:
which he explained, for instance in a later 1748 work, by setting
1305:{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n}}=\log \infty ,}
6225:"There are infinitely many primes, but, how big of an infinity?"
5062:{\displaystyle x_{k}=\sum _{n=k+1}^{\infty }{\frac {1}{p_{n}}}.}
6235:
5964:
reciprocals of primes (or indeed the sum of the reciprocals of
5960:
Another proof rewrites the expression for the sum of the first
267:
There are a variety of proofs of Euler's result, including a
3685:) for the partial sums (convergence is all we really need)
5157:
prime factors (counting multiplicities) only from the set
5999:, on the convergent sum of reciprocals of the twin primes
1752:). We will now derive an upper and a lower estimate for
5227:
4698:
2716:{\displaystyle i=(p_{1}p_{2}\cdots p_{s})\cdot b^{2},}
2463:
The proof rests upon the following four inequalities:
2408:, the estimates (2) and (3) cannot both hold, because
1940:{\displaystyle |M_{x}|\leq 2^{k}{\sqrt {x}}\qquad (2)}
680:
Here the product is taken over the set of all primes.
294:
284:
6611:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
6601:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
5767:
5663:
5630:
5394:
5327:
5288:
5163:
5143:
5103:
5077:
4998:
4731:
4649:
4527:
4406:
4304:
4298:
and taking the natural logarithm of both sides gives
3990:
3691:
3425:
3386:
3347:
3279:
3000:
2905:
2733:
2651:
2481:
2344:
2196:
2038:
1886:
1610:
1424:
1329:
1255:
723:
530:
417:
277:
103:
5692:
One proof is by induction: The first partial sum is
42:
without a subscript base should be interpreted as a
6679:
6623:
6558:
6527:
6520:
6490:
6459:
6452:
6426:
6338:
6282:
6273:
4971:. This shows that the series on the left diverges.
2391:{\displaystyle {\frac {x}{2}}<|M_{x}|\qquad (3)}
522:" to show the existence of infinitely many primes.
6186:
5911:
5669:
5649:
5610:
5378:
5313:
5274:
5213:
5149:
5129:
5089:
5061:
4959:
4715:
4606:
4467:
4382:
4272:
3973:
3638:
3400:
3368:
3329:
3265:
2986:
2888:
2715:
2625:
2390:
2328:
2127:
1939:
1780:, these bounds will turn out to be contradictory.
1680:
1554:
1522:
1408:
1304:
1230:
670:
508:
376:
234:
2456:. In the following, a sum or product taken over
18:The sum of the reciprocals of the primes diverges
6027:Commentarii Academiae Scientiarum Petropolitanae
4529:
2641:is even) or 1 (the corresponding power of prime
6095:Niven, Ivan, "A Proof of the Divergence of Σ 1/
3337:is one of the summands in the expanded product
3984:Combining all these inequalities, we see that
6247:
5275:{\textstyle \sum _{i=0}^{\infty }(x_{k})^{i}}
3412:when multiplied out. The inequality follows.
2441:Proof that the series exhibits log-log growth
8:
6694:Hypergeometric function of a matrix argument
6074:"Ein Beitrag zur analytischen Zahlentheorie"
5208:
5164:
4627:(somewhat analogous to the much more famous
2063:
2039:
6550:1 + 1/2 + 1/3 + ... (Riemann zeta function)
6056:Introduction to Infinite Analysis. Volume I
5379:{\displaystyle 1+j(p_{1}p_{2}\cdots p_{k})}
5214:{\displaystyle \{p_{k+1},p_{k+2},\cdots \}}
715:as well as the sum of a converging series:
687:. The product above is a reflection of the
6524:
6456:
6279:
6254:
6240:
6232:
3330:{\displaystyle 1/(p_{1}p_{2}\cdots p_{s})}
2637:s are 0 (the corresponding power of prime
1987:are all divisible by a prime greater than
1849:can show up (with exponent 1) in the
1555:Erdős's proof by upper and lower estimates
6606:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
6058:]. Lausanne: Bousquet. p. 228, ex. 1.
5899:
5885:
5877:
5874:
5856:
5844:
5837:
5822:
5810:
5807:
5790:
5781:
5768:
5766:
5662:
5635:
5629:
5592:
5586:
5575:
5562:
5549:
5539:
5523:
5508:
5495:
5485:
5463:
5457:
5446:
5433:
5423:
5410:
5399:
5393:
5367:
5354:
5344:
5326:
5293:
5287:
5266:
5256:
5243:
5232:
5226:
5190:
5171:
5162:
5142:
5121:
5111:
5102:
5076:
5048:
5039:
5033:
5016:
5003:
4997:
4923:
4917:
4906:
4850:
4844:
4833:
4811:
4802:
4796:
4785:
4766:
4757:
4751:
4740:
4732:
4730:
4697:
4654:
4648:
4565:
4553:
4532:
4526:
4454:
4448:
4437:
4428:
4422:
4411:
4405:
4370:
4358:
4341:
4303:
4251:
4239:
4214:
4190:
4168:
4154:
4136:
4127:
4121:
4110:
4091:
4068:
4044:
4038:
4027:
3991:
3989:
3957:
3941:
3929:
3916:
3890:
3881:
3880:
3876:
3863:
3854:
3844:
3843:
3839:
3816:
3804:
3788:
3776:
3769:
3762:
3751:
3726:
3717:
3711:
3700:
3692:
3690:
3622:
3616:
3605:
3579:
3578:
3574:
3572:
3571:
3550:
3538:
3533:
3526:
3519:
3508:
3479:
3467:
3462:
3426:
3424:
3390:
3385:
3360:
3351:
3346:
3318:
3305:
3295:
3283:
3278:
3241:
3228:
3218:
3208:
3180:
3171:
3152:
3143:
3127:
3118:
3094:
3085:
3058:
3049:
3025:
3016:
3001:
2999:
2973:
2964:
2952:
2939:
2929:
2919:
2906:
2904:
2858:
2849:
2843:
2832:
2800:
2777:
2755:
2749:
2738:
2732:
2704:
2688:
2675:
2665:
2650:
2612:
2607:
2597:
2592:
2587:
2582:
2567:
2562:
2552:
2547:
2542:
2537:
2522:
2517:
2507:
2502:
2497:
2492:
2480:
2373:
2367:
2358:
2345:
2343:
2318:
2309:
2303:
2286:
2274:
2262:
2253:
2247:
2230:
2218:
2212:
2203:
2195:
2113:
2103:
2086:
2073:
2037:
1920:
1914:
1902:
1896:
1887:
1885:
1658:
1647:
1638:
1632:
1615:
1609:
1483:
1470:
1457:
1444:
1431:
1423:
1392:
1386:
1380:
1369:
1340:
1328:
1277:
1271:
1260:
1254:
1186:
1170:
1154:
1124:
1115:
1109:
1095:
1084:
1075:
1069:
1055:
1044:
1035:
1029:
1015:
1002:
996:
962:
949:
937:
924:
911:
900:
871:
848:
821:
805:
799:
779:
760:
754:
743:
724:
722:
656:
640:
634:
608:
599:
586:
569:
552:
546:
535:
529:
484:
471:
458:
439:
433:
422:
416:
363:
357:
308:
288:
282:
276:
210:
197:
184:
171:
158:
145:
132:
119:
112:
108:
102:
1734:which are a product of powers of primes
683:Such infinite products are today called
72:
6014:
5221:. It follows that the geometric series
2066:
254:there are infinitely many prime numbers
3408:is represented in one of the terms of
5988:that there are infinitely many primes
4979:The following proof is modified from
7:
6119:"On the series of prime reciprocals"
2401:This produces a contradiction: when
6571:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
6050:Introductio in analysin infinitorum
5941:is odd; since this sum also has an
4975:Geometric and harmonic-series proof
1880:. This gives us the upper estimate
518:He had already used the following "
5605:
5587:
5458:
5411:
5244:
5034:
4950:
4918:
4845:
4797:
4752:
4539:
4423:
2304:
2248:
2104:
1633:
1581:th prime number. Assume that the
1514:
1418:This allowed him to conclude that
1381:
1296:
1272:
755:
547:
503:
434:
283:
271:for the partial sums stating that
229:
25:
6689:Generalized hypergeometric series
6101:The American Mathematical Monthly
5386:always satisfies this criterion,
2473:fundamental theorem of arithmetic
1865:. Furthermore, there are at most
1861:different possibilities for
1585:of the reciprocals of the primes
689:fundamental theorem of arithmetic
6727:
6726:
6699:Lauricella hypergeometric series
6417:
2137:Since the number of integers in
6709:Riemann's differential equation
5968:set of primes) in terms of the
4696:
2378:
1927:
1668:
1321:in the Taylor series expansion
1247:. Then he invoked the relation
5514:
5478:
5430:
5416:
5373:
5337:
5263:
5249:
5118:
5104:
4635:Proof from Dusart's inequality
4536:
4329:
4317:
4013:
4001:
3448:
3436:
3324:
3288:
2694:
2658:
2385:
2379:
2374:
2359:
2275:
2254:
2219:
2204:
1934:
1928:
1903:
1888:
1675:
1669:
345:
333:
252:'s 3rd-century-BC result that
1:
6704:Modular hypergeometric series
6545:1/4 + 1/16 + 1/64 + 1/256 + ⋯
5314:{\displaystyle p_{n},n\leq k}
4969:integral test for convergence
1592:Then there exists a smallest
260:'s 14th-century proof of the
6758:Theorems about prime numbers
4521:; in fact it turns out that
3681:. The upper bound (using a
6714:Theta hypergeometric series
6189:Euler: The Master of Us All
6105:Euler: The Master of Us All
6003:List of sums of reciprocals
5650:{\displaystyle x_{k}\geq 1}
5130:{\displaystyle (x_{k})^{i}}
3416:The upper estimate for the
2021:which are divisible by the
46:, also commonly written as
6774:
6596:Infinite arithmetic series
6540:1/2 + 1/4 + 1/8 + 1/16 + ⋯
6535:1/2 − 1/4 + 1/8 − 1/16 + ⋯
4634:
3376:is one of the summands of
1717:by any prime greater than
6722:
6415:
5992:Small set (combinatorics)
5620:by the divergence of the
4629:Euler–Mascheroni constant
1876:possible values for
248:in 1737, and strengthens
6145:The Mathematical Gazette
6117:Clarkson, James (1966).
5970:least common denominator
4625:Meissel–Mertens constant
2338:Using (1), this implies
2009:denote the set of those
1701:denote the set of those
398:Meissel–Mertens constant
384:for all natural numbers
6427:Properties of sequences
5090:{\displaystyle i\geq 0}
3369:{\displaystyle 1/b^{2}}
2899:To see this, note that
2727:s gives the inequality
2467:Every positive integer
1813:with positive integers
1690:For a positive integer
6290:Arithmetic progression
5913:
5671:
5651:
5612:
5591:
5462:
5415:
5380:
5315:
5276:
5248:
5215:
5151:
5131:
5091:
5063:
5038:
4961:
4922:
4849:
4801:
4756:
4717:
4608:
4481:), the above constant
4469:
4427:
4384:
4274:
4126:
4043:
3975:
3767:
3716:
3664:, which holds for all
3640:
3621:
3524:
3402:
3370:
3331:
3267:
2988:
2890:
2848:
2754:
2717:
2627:
2392:
2330:
2308:
2252:
2129:
2108:
1941:
1682:
1637:
1561:proof by contradiction
1524:
1410:
1385:
1306:
1276:
1232:
759:
672:
551:
510:
438:
378:
236:
78:
6681:Hypergeometric series
6295:Geometric progression
6126:Proc. Amer. Math. Soc
6079:J. Reine Angew. Math.
5914:
5708:, which has the form
5672:
5652:
5613:
5571:
5442:
5395:
5381:
5316:
5277:
5228:
5216:
5152:
5132:
5092:
5064:
5012:
4962:
4902:
4829:
4781:
4736:
4718:
4609:
4470:
4407:
4385:
4275:
4106:
4023:
3976:
3747:
3696:
3641:
3601:
3504:
3403:
3371:
3332:
3268:
2989:
2891:
2828:
2734:
2718:
2628:
2393:
2331:
2282:
2226:
2130:
2082:
1942:
1724:(or equivalently all
1683:
1611:
1525:
1411:
1365:
1307:
1256:
1240:for a fixed constant
1233:
739:
699:
673:
531:
511:
418:
379:
237:
76:
6661:Trigonometric series
6453:Properties of series
6300:Harmonic progression
5765:
5728:th partial sum (for
5661:
5628:
5392:
5325:
5286:
5225:
5161:
5141:
5101:
5075:
4996:
4729:
4647:
4525:
4404:
4302:
4282:Dividing through by
3988:
3689:
3662:exponential function
3423:
3384:
3345:
3277:
2998:
2903:
2731:
2649:
2479:
2342:
2194:
2036:
1884:
1857:, there are at most
1608:
1422:
1327:
1253:
721:
528:
415:
275:
101:
6753:Mathematical series
6641:Formal power series
6223:Caldwell, Chris K.
6157:10.1017/mag.2014.16
5097:, the expansion of
4641:Dusart's inequality
4501:can be improved to
3648:The lower estimate
3549:
3478:
3401:{\displaystyle 1/i}
2619:
2574:
2529:
2177:(actually zero for
1959: − |
1851:prime factorization
404:The harmonic series
244:This was proved by
6439:Monotonic function
6358:Fibonacci sequence
5909:
5667:
5647:
5624:. This shows that
5608:
5376:
5311:
5272:
5211:
5147:
5127:
5087:
5059:
4957:
4955:
4713:
4702:
4604:
4564:
4543:
4465:
4380:
4369:
4270:
4268:
4250:
4179:
4079:
3971:
3969:
3899:
3837:
3636:
3634:
3590:
3569:
3529:
3458:
3398:
3366:
3327:
3263:
3261:
2984:
2886:
2788:
2713:
2623:
2578:
2533:
2488:
2388:
2326:
2125:
1937:
1800:can be written as
1767:number of elements
1678:
1520:
1406:
1302:
1228:
1226:
1114:
1074:
1034:
1001:
905:
853:
804:
668:
639:
574:
506:
374:
307:
304:
293:
232:
118:
79:
6740:
6739:
6671:Generating series
6619:
6618:
6591:1 − 2 + 4 − 8 + ⋯
6586:1 + 2 + 4 + 8 + ⋯
6581:1 − 2 + 3 − 4 + ⋯
6576:1 + 2 + 3 + 4 + ⋯
6566:1 + 1 + 1 + 1 + ⋯
6516:
6515:
6444:Periodic sequence
6413:
6412:
6398:Triangular number
6388:Pentagonal number
6368:Heptagonal number
6353:Complete sequence
6275:Integer sequences
5907:
5906:
5903:
5894:
5893:
5888:
5880:
5869:
5847:
5840:
5813:
5802:
5776:
5775:
5772:
5670:{\displaystyle k}
5600:
5569:
5518:
5150:{\displaystyle i}
5054:
4981:James A. Clarkson
4945:
4890:
4817:
4772:
4701:
4573:
4549:
4528:
4463:
4443:
4392:as desired.
4378:
4354:
4349:
4259:
4235:
4222:
4198:
4164:
4162:
4142:
4099:
4064:
4052:
3965:
3952:
3949:
3924:
3896:
3874:
3871:
3827:
3824:
3799:
3796:
3770:
3768:
3732:
3630:
3587:
3563:
3527:
3525:
3492:
3418:natural logarithm
3248:
3186:
3158:
3133:
3100:
3064:
3031:
2979:
2959:
2914:
2864:
2808:
2773:
2763:
2353:
2324:
1925:
1829:. Since only the
1776:. For large
1666:
1653:
1536:is asymptotic to
1491:
1478:
1465:
1452:
1439:
1401:
1356:
1285:
1194:
1178:
1162:
1130:
1105:
1103:
1090:
1065:
1063:
1050:
1025:
1023:
1010:
992:
969:
944:
919:
896:
879:
844:
831:
795:
768:
666:
630:
614:
594:
565:
560:
492:
479:
466:
447:
390:natural logarithm
372:
316:
305:
291:
278:
218:
205:
192:
179:
166:
153:
140:
127:
115:
104:
44:natural logarithm
16:(Redirected from
6765:
6730:
6729:
6656:Dirichlet series
6525:
6457:
6421:
6393:Polygonal number
6373:Hexagonal number
6346:
6280:
6256:
6249:
6242:
6233:
6228:
6212:
6192:
6169:
6168:
6140:
6134:
6133:
6123:
6114:
6108:
6098:
6093:
6087:
6086:
6066:
6060:
6059:
6041:
6035:
6034:
6019:
5986:Euclid's theorem
5963:
5956:
5954:
5953:
5950:
5947:
5940:
5928:
5918:
5916:
5915:
5910:
5908:
5904:
5901:
5900:
5895:
5891:
5890:
5889:
5886:
5881:
5878:
5875:
5870:
5868:
5867:
5866:
5848:
5845:
5842:
5841:
5838:
5833:
5832:
5814:
5811:
5808:
5803:
5801:
5800:
5782:
5777:
5773:
5770:
5769:
5758:
5750:
5748:
5747:
5744:
5741:
5734:
5727:
5723:
5721:
5720:
5717:
5714:
5707:
5705:
5704:
5701:
5698:
5676:
5674:
5673:
5668:
5656:
5654:
5653:
5648:
5640:
5639:
5617:
5615:
5614:
5609:
5601:
5593:
5590:
5585:
5570:
5568:
5567:
5566:
5554:
5553:
5544:
5543:
5524:
5519:
5517:
5513:
5512:
5500:
5499:
5490:
5489:
5464:
5461:
5456:
5438:
5437:
5428:
5427:
5414:
5409:
5385:
5383:
5382:
5377:
5372:
5371:
5359:
5358:
5349:
5348:
5320:
5318:
5317:
5312:
5298:
5297:
5281:
5279:
5278:
5273:
5271:
5270:
5261:
5260:
5247:
5242:
5220:
5218:
5217:
5212:
5201:
5200:
5182:
5181:
5156:
5154:
5153:
5148:
5136:
5134:
5133:
5128:
5126:
5125:
5116:
5115:
5096:
5094:
5093:
5088:
5068:
5066:
5065:
5060:
5055:
5053:
5052:
5040:
5037:
5032:
5008:
5007:
4966:
4964:
4963:
4958:
4956:
4946:
4944:
4924:
4921:
4916:
4895:
4891:
4889:
4851:
4848:
4843:
4822:
4818:
4816:
4815:
4803:
4800:
4795:
4773:
4771:
4770:
4758:
4755:
4750:
4722:
4720:
4719:
4714:
4703:
4699:
4659:
4658:
4622:
4613:
4611:
4610:
4605:
4597:
4593:
4574:
4566:
4563:
4542:
4520:
4518:
4516:
4515:
4512:
4509:
4500:
4498:
4496:
4495:
4492:
4489:
4474:
4472:
4471:
4466:
4464:
4459:
4458:
4449:
4444:
4442:
4441:
4429:
4426:
4421:
4389:
4387:
4386:
4381:
4379:
4371:
4368:
4350:
4342:
4297:
4295:
4294:
4291:
4288:
4279:
4277:
4276:
4271:
4269:
4265:
4261:
4260:
4252:
4249:
4223:
4215:
4207:
4203:
4199:
4191:
4178:
4163:
4155:
4147:
4143:
4141:
4140:
4128:
4125:
4120:
4105:
4101:
4100:
4092:
4078:
4057:
4053:
4045:
4042:
4037:
3980:
3978:
3977:
3972:
3970:
3966:
3958:
3953:
3951:
3950:
3942:
3930:
3925:
3917:
3903:
3898:
3897:
3895:
3894:
3882:
3875:
3873:
3872:
3864:
3859:
3858:
3845:
3838:
3833:
3829:
3828:
3826:
3825:
3817:
3805:
3800:
3798:
3797:
3789:
3777:
3766:
3761:
3733:
3731:
3730:
3718:
3715:
3710:
3680:
3670:
3659:
3645:
3643:
3642:
3637:
3635:
3631:
3623:
3620:
3615:
3594:
3589:
3588:
3580:
3573:
3570:
3565:
3564:
3559:
3551:
3548:
3537:
3523:
3518:
3497:
3493:
3488:
3480:
3477:
3466:
3411:
3407:
3405:
3404:
3399:
3394:
3380:, every summand
3379:
3375:
3373:
3372:
3367:
3365:
3364:
3355:
3340:
3336:
3334:
3333:
3328:
3323:
3322:
3310:
3309:
3300:
3299:
3287:
3272:
3270:
3269:
3264:
3262:
3249:
3247:
3246:
3245:
3233:
3232:
3223:
3222:
3209:
3201:
3191:
3187:
3185:
3184:
3172:
3163:
3159:
3157:
3156:
3144:
3138:
3134:
3132:
3131:
3119:
3106:
3102:
3101:
3099:
3098:
3086:
3070:
3066:
3065:
3063:
3062:
3050:
3037:
3033:
3032:
3030:
3029:
3017:
2993:
2991:
2990:
2985:
2980:
2978:
2977:
2965:
2960:
2958:
2957:
2956:
2944:
2943:
2934:
2933:
2920:
2915:
2907:
2895:
2893:
2892:
2887:
2870:
2866:
2865:
2863:
2862:
2850:
2847:
2842:
2819:
2815:
2814:
2810:
2809:
2801:
2787:
2764:
2756:
2753:
2748:
2726:
2722:
2720:
2719:
2714:
2709:
2708:
2693:
2692:
2680:
2679:
2670:
2669:
2644:
2640:
2632:
2630:
2629:
2624:
2618:
2617:
2616:
2611:
2602:
2601:
2596:
2586:
2573:
2572:
2571:
2566:
2557:
2556:
2551:
2541:
2528:
2527:
2526:
2521:
2512:
2511:
2506:
2496:
2470:
2459:
2451:
2436:
2435:
2434:
2426:
2424:
2423:
2420:
2417:
2407:
2397:
2395:
2394:
2389:
2377:
2372:
2371:
2362:
2354:
2346:
2335:
2333:
2332:
2327:
2325:
2323:
2322:
2310:
2307:
2302:
2278:
2273:
2272:
2257:
2251:
2246:
2222:
2217:
2216:
2207:
2189:
2176:
2175:
2173:
2172:
2164:
2161:
2151:
2134:
2132:
2131:
2126:
2124:
2123:
2107:
2102:
2078:
2077:
2031:
2024:
2020:
2012:
2008:
1993:
1986:
1969:
1967:
1946:
1944:
1943:
1938:
1926:
1921:
1919:
1918:
1906:
1901:
1900:
1891:
1879:
1875:
1874:
1873:
1864:
1860:
1856:
1848:
1832:
1824:
1820:
1816:
1812:
1799:
1792:
1779:
1775:
1764:
1762:
1751:
1733:
1723:
1712:
1704:
1700:
1693:
1687:
1685:
1684:
1679:
1667:
1659:
1654:
1652:
1651:
1639:
1636:
1631:
1601:
1580:
1576:
1546:
1542:
1535:
1529:
1527:
1526:
1521:
1492:
1484:
1479:
1471:
1466:
1458:
1453:
1445:
1440:
1432:
1415:
1413:
1412:
1407:
1402:
1397:
1396:
1387:
1384:
1379:
1361:
1357:
1355:
1341:
1320:
1311:
1309:
1308:
1303:
1286:
1278:
1275:
1270:
1246:
1237:
1235:
1234:
1229:
1227:
1208:
1195:
1187:
1179:
1171:
1163:
1155:
1141:
1131:
1129:
1128:
1116:
1113:
1104:
1096:
1091:
1089:
1088:
1076:
1073:
1064:
1056:
1051:
1049:
1048:
1036:
1033:
1024:
1016:
1011:
1003:
1000:
985:
981:
977:
970:
968:
967:
966:
950:
945:
943:
942:
941:
925:
920:
912:
904:
889:
885:
881:
880:
872:
852:
837:
833:
832:
830:
829:
828:
806:
803:
780:
774:
770:
769:
761:
758:
753:
714:
677:
675:
674:
669:
667:
665:
664:
663:
641:
638:
626:
622:
615:
613:
612:
600:
595:
587:
573:
561:
553:
550:
545:
515:
513:
512:
507:
493:
485:
480:
472:
467:
459:
448:
440:
437:
432:
395:
387:
383:
381:
380:
375:
373:
368:
367:
358:
317:
309:
306:
292:
289:
241:
239:
238:
233:
219:
211:
206:
198:
193:
185:
180:
172:
167:
159:
154:
146:
141:
133:
128:
120:
117:
116:
113:
67:
53:
41:
21:
6773:
6772:
6768:
6767:
6766:
6764:
6763:
6762:
6743:
6742:
6741:
6736:
6718:
6675:
6624:Kinds of series
6615:
6554:
6521:Explicit series
6512:
6486:
6448:
6434:Cauchy sequence
6422:
6409:
6363:Figurate number
6340:
6334:
6325:Powers of three
6269:
6260:
6222:
6219:
6209:
6183:Dunham, William
6181:
6173:
6172:
6142:
6141:
6137:
6121:
6116:
6115:
6111:
6096:
6094:
6090:
6068:
6067:
6063:
6045:Euler, Leonhard
6043:
6042:
6038:
6023:Euler, Leonhard
6021:
6020:
6016:
6011:
5982:
5961:
5951:
5948:
5945:
5944:
5942:
5939:
5930:
5922:
5876:
5852:
5843:
5818:
5809:
5786:
5763:
5762:
5752:
5745:
5742:
5739:
5738:
5736:
5735:) has the form
5729:
5725:
5718:
5715:
5712:
5711:
5709:
5702:
5699:
5696:
5695:
5693:
5683:
5659:
5658:
5631:
5626:
5625:
5622:harmonic series
5558:
5545:
5535:
5528:
5504:
5491:
5481:
5468:
5429:
5419:
5390:
5389:
5363:
5350:
5340:
5323:
5322:
5289:
5284:
5283:
5262:
5252:
5223:
5222:
5186:
5167:
5159:
5158:
5139:
5138:
5117:
5107:
5099:
5098:
5073:
5072:
5044:
4999:
4994:
4993:
4977:
4954:
4953:
4928:
4893:
4892:
4855:
4820:
4819:
4807:
4774:
4762:
4727:
4726:
4650:
4645:
4644:
4637:
4617:
4548:
4544:
4523:
4522:
4513:
4510:
4507:
4506:
4504:
4502:
4493:
4490:
4487:
4486:
4484:
4482:
4450:
4433:
4402:
4401:
4300:
4299:
4292:
4289:
4286:
4285:
4283:
4267:
4266:
4234:
4230:
4205:
4204:
4186:
4145:
4144:
4132:
4084:
4080:
4055:
4054:
4016:
3986:
3985:
3968:
3967:
3934:
3901:
3900:
3886:
3850:
3849:
3809:
3781:
3775:
3771:
3734:
3722:
3687:
3686:
3683:telescoping sum
3675:
3665:
3649:
3633:
3632:
3592:
3591:
3552:
3528:
3495:
3494:
3481:
3451:
3421:
3420:
3409:
3382:
3381:
3377:
3356:
3343:
3342:
3338:
3314:
3301:
3291:
3275:
3274:
3260:
3259:
3237:
3224:
3214:
3213:
3199:
3198:
3176:
3167:
3148:
3139:
3123:
3114:
3107:
3090:
3078:
3074:
3054:
3042:
3038:
3021:
3009:
3005:
2996:
2995:
2969:
2948:
2935:
2925:
2924:
2901:
2900:
2854:
2827:
2823:
2793:
2789:
2772:
2768:
2729:
2728:
2724:
2700:
2684:
2671:
2661:
2647:
2646:
2642:
2638:
2606:
2591:
2561:
2546:
2516:
2501:
2477:
2476:
2468:
2457:
2446:
2443:
2430:
2428:
2421:
2418:
2413:
2412:
2410:
2409:
2402:
2363:
2340:
2339:
2314:
2258:
2208:
2192:
2191:
2183:
2178:
2170:
2165:
2162:
2157:
2156:
2154:
2153:
2150:
2138:
2109:
2069:
2034:
2033:
2030:
2026:
2022:
2014:
2010:
2007:
1995:
1992:
1988:
1984:
1974:
1970:numbers in the
1965:
1960:
1955:
1910:
1892:
1882:
1881:
1877:
1869:
1867:
1866:
1862:
1858:
1854:
1846:
1840:
1834:
1830:
1822:
1818:
1814:
1801:
1798:
1794:
1790:
1777:
1774:
1770:
1760:
1755:
1753:
1750:
1740:
1735:
1725:
1722:
1718:
1706:
1702:
1699:
1695:
1691:
1643:
1606:
1605:
1599:
1578:
1575:
1571:
1557:
1544:
1537:
1533:
1420:
1419:
1388:
1345:
1336:
1325:
1324:
1315:
1251:
1250:
1241:
1225:
1224:
1206:
1205:
1139:
1138:
1120:
1080:
1040:
983:
982:
958:
954:
933:
929:
910:
906:
887:
886:
864:
860:
817:
810:
794:
790:
775:
738:
734:
719:
718:
709:
702:
697:
652:
645:
604:
579:
575:
526:
525:
520:product formula
413:
412:
410:harmonic series
406:
393:
385:
359:
273:
272:
99:
98:
71:
70:
69:
61:
55:
47:
35:
28:
23:
22:
15:
12:
11:
5:
6771:
6769:
6761:
6760:
6755:
6745:
6744:
6738:
6737:
6735:
6734:
6723:
6720:
6719:
6717:
6716:
6711:
6706:
6701:
6696:
6691:
6685:
6683:
6677:
6676:
6674:
6673:
6668:
6666:Fourier series
6663:
6658:
6653:
6651:Puiseux series
6648:
6646:Laurent series
6643:
6638:
6633:
6627:
6625:
6621:
6620:
6617:
6616:
6614:
6613:
6608:
6603:
6598:
6593:
6588:
6583:
6578:
6573:
6568:
6562:
6560:
6556:
6555:
6553:
6552:
6547:
6542:
6537:
6531:
6529:
6522:
6518:
6517:
6514:
6513:
6511:
6510:
6505:
6500:
6494:
6492:
6488:
6487:
6485:
6484:
6479:
6474:
6469:
6463:
6461:
6454:
6450:
6449:
6447:
6446:
6441:
6436:
6430:
6428:
6424:
6423:
6416:
6414:
6411:
6410:
6408:
6407:
6406:
6405:
6395:
6390:
6385:
6380:
6375:
6370:
6365:
6360:
6355:
6349:
6347:
6336:
6335:
6333:
6332:
6327:
6322:
6317:
6312:
6307:
6302:
6297:
6292:
6286:
6284:
6277:
6271:
6270:
6261:
6259:
6258:
6251:
6244:
6236:
6230:
6229:
6218:
6217:External links
6215:
6214:
6213:
6207:
6171:
6170:
6135:
6109:
6088:
6061:
6052:. Tomus Primus
6036:
6013:
6012:
6010:
6007:
6006:
6005:
6000:
5997:Brun's theorem
5994:
5989:
5981:
5978:
5934:
5898:
5884:
5873:
5865:
5862:
5859:
5855:
5851:
5836:
5831:
5828:
5825:
5821:
5817:
5806:
5799:
5796:
5793:
5789:
5785:
5780:
5682:
5679:
5666:
5646:
5643:
5638:
5634:
5607:
5604:
5599:
5596:
5589:
5584:
5581:
5578:
5574:
5565:
5561:
5557:
5552:
5548:
5542:
5538:
5534:
5531:
5527:
5522:
5516:
5511:
5507:
5503:
5498:
5494:
5488:
5484:
5480:
5477:
5474:
5471:
5467:
5460:
5455:
5452:
5449:
5445:
5441:
5436:
5432:
5426:
5422:
5418:
5413:
5408:
5405:
5402:
5398:
5375:
5370:
5366:
5362:
5357:
5353:
5347:
5343:
5339:
5336:
5333:
5330:
5310:
5307:
5304:
5301:
5296:
5292:
5269:
5265:
5259:
5255:
5251:
5246:
5241:
5238:
5235:
5231:
5210:
5207:
5204:
5199:
5196:
5193:
5189:
5185:
5180:
5177:
5174:
5170:
5166:
5146:
5124:
5120:
5114:
5110:
5106:
5086:
5083:
5080:
5058:
5051:
5047:
5043:
5036:
5031:
5028:
5025:
5022:
5019:
5015:
5011:
5006:
5002:
4976:
4973:
4952:
4949:
4943:
4940:
4937:
4934:
4931:
4927:
4920:
4915:
4912:
4909:
4905:
4901:
4898:
4896:
4894:
4888:
4885:
4882:
4879:
4876:
4873:
4870:
4867:
4864:
4861:
4858:
4854:
4847:
4842:
4839:
4836:
4832:
4828:
4825:
4823:
4821:
4814:
4810:
4806:
4799:
4794:
4791:
4788:
4784:
4780:
4777:
4775:
4769:
4765:
4761:
4754:
4749:
4746:
4743:
4739:
4735:
4734:
4712:
4709:
4706:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4657:
4653:
4636:
4633:
4603:
4600:
4596:
4592:
4589:
4586:
4583:
4580:
4577:
4572:
4569:
4562:
4559:
4556:
4552:
4547:
4541:
4538:
4535:
4531:
4462:
4457:
4453:
4447:
4440:
4436:
4432:
4425:
4420:
4417:
4414:
4410:
4377:
4374:
4367:
4364:
4361:
4357:
4353:
4348:
4345:
4340:
4337:
4334:
4331:
4328:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4264:
4258:
4255:
4248:
4245:
4242:
4238:
4233:
4229:
4226:
4221:
4218:
4213:
4210:
4208:
4206:
4202:
4197:
4194:
4189:
4185:
4182:
4177:
4174:
4171:
4167:
4161:
4158:
4153:
4150:
4148:
4146:
4139:
4135:
4131:
4124:
4119:
4116:
4113:
4109:
4104:
4098:
4095:
4090:
4087:
4083:
4077:
4074:
4071:
4067:
4063:
4060:
4058:
4056:
4051:
4048:
4041:
4036:
4033:
4030:
4026:
4022:
4019:
4017:
4015:
4012:
4009:
4006:
4003:
4000:
3997:
3994:
3993:
3982:
3981:
3964:
3961:
3956:
3948:
3945:
3940:
3937:
3933:
3928:
3923:
3920:
3915:
3912:
3909:
3906:
3904:
3902:
3893:
3889:
3885:
3879:
3870:
3867:
3862:
3857:
3853:
3848:
3842:
3836:
3832:
3823:
3820:
3815:
3812:
3808:
3803:
3795:
3792:
3787:
3784:
3780:
3774:
3765:
3760:
3757:
3754:
3750:
3746:
3743:
3740:
3737:
3735:
3729:
3725:
3721:
3714:
3709:
3706:
3703:
3699:
3695:
3694:
3672:
3646:
3629:
3626:
3619:
3614:
3611:
3608:
3604:
3600:
3597:
3595:
3593:
3586:
3583:
3577:
3568:
3562:
3558:
3555:
3547:
3544:
3541:
3536:
3532:
3522:
3517:
3514:
3511:
3507:
3503:
3500:
3498:
3496:
3491:
3487:
3484:
3476:
3473:
3470:
3465:
3461:
3457:
3454:
3452:
3450:
3447:
3444:
3441:
3438:
3435:
3432:
3429:
3428:
3397:
3393:
3389:
3363:
3359:
3354:
3350:
3326:
3321:
3317:
3313:
3308:
3304:
3298:
3294:
3290:
3286:
3282:
3258:
3255:
3252:
3244:
3240:
3236:
3231:
3227:
3221:
3217:
3212:
3207:
3204:
3202:
3200:
3197:
3194:
3190:
3183:
3179:
3175:
3170:
3166:
3162:
3155:
3151:
3147:
3142:
3137:
3130:
3126:
3122:
3117:
3113:
3110:
3108:
3105:
3097:
3093:
3089:
3084:
3081:
3077:
3073:
3069:
3061:
3057:
3053:
3048:
3045:
3041:
3036:
3028:
3024:
3020:
3015:
3012:
3008:
3004:
3003:
2983:
2976:
2972:
2968:
2963:
2955:
2951:
2947:
2942:
2938:
2932:
2928:
2923:
2918:
2913:
2910:
2897:
2896:
2885:
2882:
2879:
2876:
2873:
2869:
2861:
2857:
2853:
2846:
2841:
2838:
2835:
2831:
2826:
2822:
2818:
2813:
2807:
2804:
2799:
2796:
2792:
2786:
2783:
2780:
2776:
2771:
2767:
2762:
2759:
2752:
2747:
2744:
2741:
2737:
2712:
2707:
2703:
2699:
2696:
2691:
2687:
2683:
2678:
2674:
2668:
2664:
2660:
2657:
2654:
2622:
2615:
2610:
2605:
2600:
2595:
2590:
2585:
2581:
2577:
2570:
2565:
2560:
2555:
2550:
2545:
2540:
2536:
2532:
2525:
2520:
2515:
2510:
2505:
2500:
2495:
2491:
2487:
2484:
2442:
2439:
2399:
2398:
2387:
2384:
2381:
2376:
2370:
2366:
2361:
2357:
2352:
2349:
2336:
2321:
2317:
2313:
2306:
2301:
2298:
2295:
2292:
2289:
2285:
2281:
2277:
2271:
2268:
2265:
2261:
2256:
2250:
2245:
2242:
2239:
2236:
2233:
2229:
2225:
2221:
2215:
2211:
2206:
2202:
2199:
2181:
2168:
2142:
2135:
2122:
2119:
2116:
2112:
2106:
2101:
2098:
2095:
2092:
2089:
2085:
2081:
2076:
2072:
2068:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2044:
2041:
2028:
1999:
1990:
1982:
1972:set difference
1963:
1954:The remaining
1952:
1950:
1949:Lower estimate
1947:
1936:
1933:
1930:
1924:
1917:
1913:
1909:
1905:
1899:
1895:
1890:
1844:
1838:
1796:
1787:
1785:
1784:Upper estimate
1772:
1758:
1746:
1738:
1720:
1713:which are not
1697:
1677:
1674:
1671:
1665:
1662:
1657:
1650:
1646:
1642:
1635:
1630:
1627:
1624:
1621:
1618:
1614:
1573:
1559:The following
1556:
1553:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1490:
1487:
1482:
1477:
1474:
1469:
1464:
1461:
1456:
1451:
1448:
1443:
1438:
1435:
1430:
1427:
1405:
1400:
1395:
1391:
1383:
1378:
1375:
1372:
1368:
1364:
1360:
1354:
1351:
1348:
1344:
1339:
1335:
1332:
1301:
1298:
1295:
1292:
1289:
1284:
1281:
1274:
1269:
1266:
1263:
1259:
1223:
1220:
1217:
1214:
1211:
1209:
1207:
1204:
1201:
1198:
1193:
1190:
1185:
1182:
1177:
1174:
1169:
1166:
1161:
1158:
1153:
1150:
1147:
1144:
1142:
1140:
1137:
1134:
1127:
1123:
1119:
1112:
1108:
1102:
1099:
1094:
1087:
1083:
1079:
1072:
1068:
1062:
1059:
1054:
1047:
1043:
1039:
1032:
1028:
1022:
1019:
1014:
1009:
1006:
999:
995:
991:
988:
986:
984:
980:
976:
973:
965:
961:
957:
953:
948:
940:
936:
932:
928:
923:
918:
915:
909:
903:
899:
895:
892:
890:
888:
884:
878:
875:
870:
867:
863:
859:
856:
851:
847:
843:
840:
836:
827:
824:
820:
816:
813:
809:
802:
798:
793:
789:
786:
783:
778:
776:
773:
767:
764:
757:
752:
749:
746:
742:
737:
733:
730:
727:
726:
708:expansion for
701:
698:
696:
693:
685:Euler products
662:
659:
655:
651:
648:
644:
637:
633:
629:
625:
621:
618:
611:
607:
603:
598:
593:
590:
585:
582:
578:
572:
568:
564:
559:
556:
549:
544:
541:
538:
534:
505:
502:
499:
496:
491:
488:
483:
478:
475:
470:
465:
462:
457:
454:
451:
446:
443:
436:
431:
428:
425:
421:
405:
402:
371:
366:
362:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
326:
323:
320:
315:
312:
303:
300:
297:
287:
281:
246:Leonhard Euler
231:
228:
225:
222:
217:
214:
209:
204:
201:
196:
191:
188:
183:
178:
175:
170:
165:
162:
157:
152:
149:
144:
139:
136:
131:
126:
123:
111:
107:
57:
33:
32:
31:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6770:
6759:
6756:
6754:
6751:
6750:
6748:
6733:
6725:
6724:
6721:
6715:
6712:
6710:
6707:
6705:
6702:
6700:
6697:
6695:
6692:
6690:
6687:
6686:
6684:
6682:
6678:
6672:
6669:
6667:
6664:
6662:
6659:
6657:
6654:
6652:
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6631:Taylor series
6629:
6628:
6626:
6622:
6612:
6609:
6607:
6604:
6602:
6599:
6597:
6594:
6592:
6589:
6587:
6584:
6582:
6579:
6577:
6574:
6572:
6569:
6567:
6564:
6563:
6561:
6557:
6551:
6548:
6546:
6543:
6541:
6538:
6536:
6533:
6532:
6530:
6526:
6523:
6519:
6509:
6506:
6504:
6501:
6499:
6496:
6495:
6493:
6489:
6483:
6480:
6478:
6475:
6473:
6470:
6468:
6465:
6464:
6462:
6458:
6455:
6451:
6445:
6442:
6440:
6437:
6435:
6432:
6431:
6429:
6425:
6420:
6404:
6401:
6400:
6399:
6396:
6394:
6391:
6389:
6386:
6384:
6381:
6379:
6376:
6374:
6371:
6369:
6366:
6364:
6361:
6359:
6356:
6354:
6351:
6350:
6348:
6344:
6337:
6331:
6328:
6326:
6323:
6321:
6320:Powers of two
6318:
6316:
6313:
6311:
6308:
6306:
6305:Square number
6303:
6301:
6298:
6296:
6293:
6291:
6288:
6287:
6285:
6281:
6278:
6276:
6272:
6268:
6264:
6257:
6252:
6250:
6245:
6243:
6238:
6237:
6234:
6226:
6221:
6220:
6216:
6210:
6208:0-88385-328-0
6204:
6200:
6196:
6191:
6190:
6184:
6180:
6179:
6178:
6177:
6166:
6162:
6158:
6154:
6150:
6146:
6139:
6136:
6131:
6127:
6120:
6113:
6110:
6106:
6102:
6092:
6089:
6084:
6081:
6080:
6075:
6071:
6065:
6062:
6057:
6053:
6051:
6046:
6040:
6037:
6032:
6028:
6024:
6018:
6015:
6008:
6004:
6001:
5998:
5995:
5993:
5990:
5987:
5984:
5983:
5979:
5977:
5975:
5971:
5967:
5958:
5937:
5933:
5926:
5919:
5896:
5882:
5871:
5863:
5860:
5857:
5853:
5849:
5834:
5829:
5826:
5823:
5819:
5815:
5804:
5797:
5794:
5791:
5787:
5783:
5778:
5760:
5756:
5732:
5690:
5688:
5680:
5678:
5664:
5644:
5641:
5636:
5632:
5623:
5618:
5602:
5597:
5594:
5582:
5579:
5576:
5572:
5563:
5559:
5555:
5550:
5546:
5540:
5536:
5532:
5529:
5525:
5520:
5509:
5505:
5501:
5496:
5492:
5486:
5482:
5475:
5472:
5469:
5465:
5453:
5450:
5447:
5443:
5439:
5434:
5424:
5420:
5406:
5403:
5400:
5396:
5387:
5368:
5364:
5360:
5355:
5351:
5345:
5341:
5334:
5331:
5328:
5308:
5305:
5302:
5299:
5294:
5290:
5267:
5257:
5253:
5239:
5236:
5233:
5229:
5205:
5202:
5197:
5194:
5191:
5187:
5183:
5178:
5175:
5172:
5168:
5144:
5122:
5112:
5108:
5084:
5081:
5078:
5069:
5056:
5049:
5045:
5041:
5029:
5026:
5023:
5020:
5017:
5013:
5009:
5004:
5000:
4991:
4989:
4984:
4982:
4974:
4972:
4970:
4947:
4941:
4938:
4935:
4932:
4929:
4925:
4913:
4910:
4907:
4903:
4899:
4897:
4886:
4883:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4852:
4840:
4837:
4834:
4830:
4826:
4824:
4812:
4808:
4804:
4792:
4789:
4786:
4782:
4778:
4776:
4767:
4763:
4759:
4747:
4744:
4741:
4737:
4723:
4710:
4707:
4704:
4693:
4690:
4687:
4684:
4681:
4678:
4675:
4672:
4669:
4666:
4663:
4660:
4655:
4651:
4642:
4632:
4630:
4626:
4621:= 0.261497...
4620:
4614:
4601:
4598:
4594:
4590:
4587:
4584:
4581:
4578:
4575:
4570:
4567:
4560:
4557:
4554:
4550:
4545:
4533:
4480:
4479:Basel problem
4475:
4460:
4455:
4451:
4445:
4438:
4434:
4430:
4418:
4415:
4412:
4408:
4399:
4396:
4395:
4390:
4375:
4372:
4365:
4362:
4359:
4355:
4351:
4346:
4343:
4338:
4335:
4332:
4326:
4323:
4320:
4314:
4311:
4308:
4305:
4280:
4262:
4256:
4253:
4246:
4243:
4240:
4236:
4231:
4227:
4224:
4219:
4216:
4211:
4209:
4200:
4195:
4192:
4187:
4183:
4180:
4175:
4172:
4169:
4165:
4159:
4156:
4151:
4149:
4137:
4133:
4129:
4122:
4117:
4114:
4111:
4107:
4102:
4096:
4093:
4088:
4085:
4081:
4075:
4072:
4069:
4065:
4061:
4059:
4049:
4046:
4039:
4034:
4031:
4028:
4024:
4020:
4018:
4010:
4007:
4004:
3998:
3995:
3962:
3959:
3954:
3946:
3943:
3938:
3935:
3931:
3926:
3921:
3918:
3913:
3910:
3907:
3905:
3891:
3887:
3883:
3877:
3868:
3865:
3860:
3855:
3851:
3846:
3840:
3834:
3830:
3821:
3818:
3813:
3810:
3806:
3801:
3793:
3790:
3785:
3782:
3778:
3772:
3763:
3758:
3755:
3752:
3748:
3744:
3741:
3738:
3736:
3727:
3723:
3719:
3712:
3707:
3704:
3701:
3697:
3684:
3678:
3673:
3668:
3663:
3657:
3653:
3647:
3627:
3624:
3617:
3612:
3609:
3606:
3602:
3598:
3596:
3584:
3581:
3575:
3566:
3560:
3556:
3553:
3545:
3542:
3539:
3534:
3530:
3520:
3515:
3512:
3509:
3505:
3501:
3499:
3489:
3485:
3482:
3474:
3471:
3468:
3463:
3459:
3455:
3453:
3445:
3442:
3439:
3433:
3430:
3419:
3415:
3414:
3413:
3395:
3391:
3387:
3361:
3357:
3352:
3348:
3319:
3315:
3311:
3306:
3302:
3296:
3292:
3284:
3280:
3256:
3253:
3250:
3242:
3238:
3234:
3229:
3225:
3219:
3215:
3210:
3205:
3203:
3195:
3192:
3188:
3181:
3177:
3173:
3168:
3164:
3160:
3153:
3149:
3145:
3140:
3135:
3128:
3124:
3120:
3115:
3111:
3109:
3103:
3095:
3091:
3087:
3082:
3079:
3075:
3071:
3067:
3059:
3055:
3051:
3046:
3043:
3039:
3034:
3026:
3022:
3018:
3013:
3010:
3006:
2981:
2974:
2970:
2966:
2961:
2953:
2949:
2945:
2940:
2936:
2930:
2926:
2921:
2916:
2911:
2908:
2883:
2880:
2877:
2874:
2871:
2867:
2859:
2855:
2851:
2844:
2839:
2836:
2833:
2829:
2824:
2820:
2816:
2811:
2805:
2802:
2797:
2794:
2790:
2784:
2781:
2778:
2774:
2769:
2765:
2760:
2757:
2750:
2745:
2742:
2739:
2735:
2710:
2705:
2701:
2697:
2689:
2685:
2681:
2676:
2672:
2666:
2662:
2655:
2652:
2636:
2620:
2613:
2608:
2603:
2598:
2593:
2588:
2583:
2579:
2575:
2568:
2563:
2558:
2553:
2548:
2543:
2538:
2534:
2530:
2523:
2518:
2513:
2508:
2503:
2498:
2493:
2489:
2485:
2482:
2475:. Start with
2474:
2466:
2465:
2464:
2461:
2455:
2450:
2440:
2438:
2433:
2416:
2405:
2382:
2368:
2364:
2355:
2350:
2347:
2337:
2319:
2315:
2311:
2299:
2296:
2293:
2290:
2287:
2283:
2279:
2269:
2266:
2263:
2259:
2243:
2240:
2237:
2234:
2231:
2227:
2223:
2213:
2209:
2200:
2197:
2188:
2184:
2171:
2160:
2149:
2145:
2141:
2136:
2120:
2117:
2114:
2110:
2099:
2096:
2093:
2090:
2087:
2083:
2079:
2074:
2070:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2018:
2006:
2002:
1998:
1985:
1978:
1973:
1966:
1958:
1953:
1951:
1948:
1931:
1922:
1915:
1911:
1907:
1897:
1893:
1872:
1852:
1847:
1837:
1828:
1811:
1808:
1804:
1788:
1786:
1783:
1782:
1781:
1768:
1761:
1749:
1745:
1741:
1732:
1728:
1716:
1710:
1688:
1672:
1663:
1660:
1655:
1648:
1644:
1640:
1628:
1625:
1622:
1619:
1616:
1612:
1603:
1598:
1595:
1590:
1588:
1584:
1568:
1566:
1562:
1552:
1550:
1549:Franz Mertens
1541:
1530:
1517:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1488:
1485:
1480:
1475:
1472:
1467:
1462:
1459:
1454:
1449:
1446:
1441:
1436:
1433:
1428:
1425:
1416:
1403:
1398:
1393:
1389:
1376:
1373:
1370:
1366:
1362:
1358:
1352:
1349:
1346:
1342:
1337:
1333:
1330:
1322:
1318:
1312:
1299:
1293:
1290:
1287:
1282:
1279:
1267:
1264:
1261:
1257:
1248:
1244:
1238:
1221:
1218:
1215:
1212:
1210:
1202:
1199:
1196:
1191:
1188:
1183:
1180:
1175:
1172:
1167:
1164:
1159:
1156:
1151:
1148:
1145:
1143:
1135:
1132:
1125:
1121:
1117:
1110:
1106:
1100:
1097:
1092:
1085:
1081:
1077:
1070:
1066:
1060:
1057:
1052:
1045:
1041:
1037:
1030:
1026:
1020:
1017:
1012:
1007:
1004:
997:
993:
989:
987:
978:
974:
971:
963:
959:
955:
951:
946:
938:
934:
930:
926:
921:
916:
913:
907:
901:
897:
893:
891:
882:
876:
873:
868:
865:
861:
857:
854:
849:
845:
841:
838:
834:
825:
822:
818:
814:
811:
807:
800:
796:
791:
787:
784:
781:
777:
771:
765:
762:
750:
747:
744:
740:
735:
731:
728:
716:
713:
707:
706:Taylor series
700:Euler's proof
694:
692:
690:
686:
681:
678:
660:
657:
653:
649:
646:
642:
635:
631:
627:
623:
619:
616:
609:
605:
601:
596:
591:
588:
583:
580:
576:
570:
566:
562:
557:
554:
542:
539:
536:
532:
523:
521:
516:
500:
497:
494:
489:
486:
481:
476:
473:
468:
463:
460:
455:
452:
449:
444:
441:
429:
426:
423:
419:
411:
403:
401:
399:
391:
388:. The double
369:
364:
360:
354:
351:
348:
342:
339:
336:
330:
327:
324:
321:
318:
313:
310:
301:
298:
295:
285:
279:
270:
265:
263:
259:
258:Nicole Oresme
255:
251:
247:
242:
226:
223:
220:
215:
212:
207:
202:
199:
194:
189:
186:
181:
176:
173:
168:
163:
160:
155:
150:
147:
142:
137:
134:
129:
124:
121:
109:
105:
96:
94:
93:
90:
89:prime numbers
86:
75:
65:
60:
51:
45:
39:
30:
19:
6636:Power series
6610:
6378:Lucas number
6330:Powers of 10
6310:Cubic number
6188:
6175:
6174:
6148:
6144:
6138:
6129:
6125:
6112:
6107:, pp. 74-76.
6104:
6100:
6091:
6082:
6077:
6064:
6055:
6048:
6039:
6030:
6026:
6017:
5973:
5965:
5959:
5935:
5931:
5924:
5920:
5761:
5754:
5730:
5691:
5687:partial sums
5684:
5681:Partial sums
5619:
5388:
5321:. But since
5070:
4992:
4987:
4985:
4978:
4724:
4638:
4618:
4615:
4499:= 0.51082...
4476:
4400:
4397:
4391:
4281:
3983:
3676:
3666:
3655:
3651:
3341:. And since
2898:
2634:
2462:
2448:
2444:
2431:
2414:
2403:
2400:
2186:
2179:
2166:
2158:
2147:
2143:
2139:
2016:
2015:{1, 2, ...,
2004:
2000:
1996:
1980:
1976:
1975:{1, 2, ...,
1961:
1956:
1870:
1842:
1835:
1809:
1806:
1802:
1756:
1747:
1743:
1736:
1730:
1726:
1708:
1707:{1, 2, ...,
1689:
1604:
1591:
1569:
1558:
1539:
1531:
1417:
1323:
1316:
1313:
1249:
1242:
1239:
717:
711:
703:
682:
679:
524:
517:
407:
266:
243:
97:
82:
80:
63:
58:
49:
37:
29:
6503:Conditional
6491:Convergence
6482:Telescoping
6467:Alternating
6383:Pell number
6197:. pp.
6151:: 128–130.
6070:Mertens, F.
5751:, then the
4986:Define the
4519:= 0.4977...
2152:is at most
1827:square-free
1577:denote the
1563:comes from
290: prime
269:lower bound
114: prime
95:; that is:
85:reciprocals
83:sum of the
6747:Categories
6528:Convergent
6472:Convergent
6033:: 160–188.
6009:References
5759:st sum is
5685:While the
2633:where the
2190:), we get
1602:such that
1565:Paul Erdős
6559:Divergent
6477:Divergent
6339:Advanced
6315:Factorial
6263:Sequences
6165:123890989
5929:st prime
5850:⋅
5816:⋅
5724:. If the
5642:≥
5606:∞
5588:∞
5573:∑
5556:⋯
5502:⋯
5459:∞
5444:∑
5412:∞
5397:∑
5361:⋯
5306:≤
5245:∞
5230:∑
5206:⋯
5082:≥
5071:Then for
5035:∞
5014:∑
4990:-th tail
4951:∞
4939:
4919:∞
4904:∑
4900:≥
4884:
4878:
4863:
4846:∞
4831:∑
4827:≥
4798:∞
4783:∑
4779:≥
4753:∞
4738:∑
4708:≥
4700:for
4691:
4685:
4670:
4643:, we get
4588:
4582:
4576:−
4558:≤
4551:∑
4540:∞
4537:→
4477:(see the
4452:π
4424:∞
4409:∑
4363:≤
4356:∑
4339:
4333:−
4315:
4309:
4244:≤
4237:∑
4228:
4184:
4173:≤
4166:∏
4108:∑
4073:≤
4066:∏
4062:≤
4025:∑
3999:
3927:−
3861:−
3835:⏟
3802:−
3786:−
3749:∑
3698:∑
3654:< exp(
3603:∑
3567:⏟
3531:∫
3506:∑
3460:∫
3434:
3312:⋯
3273:That is,
3254:…
3235:⋯
3196:…
3165:⋯
3072:…
2962:⋅
2946:⋯
2878:⋅
2830:∑
2821:⋅
2782:≤
2775:∏
2766:≤
2736:∑
2698:⋅
2682:⋯
2609:β
2594:α
2576:⋯
2564:β
2549:α
2531:⋅
2519:β
2504:α
2305:∞
2284:∑
2249:∞
2228:∑
2224:≤
2201:−
2105:∞
2084:⋃
2067:∖
2055:…
2025:th prime
1908:≤
1715:divisible
1634:∞
1613:∑
1587:converges
1515:∞
1512:
1506:
1497:⋯
1382:∞
1367:∑
1350:−
1334:
1297:∞
1294:
1273:∞
1258:∑
1203:⋯
1136:⋯
1107:∑
1067:∑
1027:∑
994:∑
975:⋯
898:∑
869:−
858:
846:∑
842:−
823:−
815:−
797:∏
788:
756:∞
741:∑
732:
658:−
650:−
632:∏
620:⋯
567:∏
548:∞
533:∑
504:∞
498:⋯
435:∞
420:∑
361:π
355:
349:−
331:
325:
319:≥
299:≤
280:∑
230:∞
224:⋯
106:∑
6732:Category
6498:Absolute
6185:(1999).
6085:: 46–62.
6072:(1874).
6047:(1748).
5980:See also
5657:for all
3660:for the
2447:log log
1853:of
1821:, where
1594:positive
1538:log log
92:diverges
6508:Uniform
6176:Sources
5955:
5943:
5921:as the
5749:
5737:
5722:
5710:
5706:
5694:
4967:by the
4623:is the
4517:
4505:
4497:
4485:
4296:
4284:
2429:√
2425:
2411:
2174:
2155:
2032:. Then
1868:√
1841:, ...,
1833:primes
1597:integer
394:log log
87:of all
27:Theorem
6460:Series
6267:series
6205:
6163:
6132:: 541.
4616:where
4398:Using
4394:Q.E.D.
3669:> 0
1994:. Let
1968:|
1789:Every
1765:, the
1763:|
1754:|
1694:, let
1245:< 1
695:Proofs
250:Euclid
6403:array
6283:Basic
6199:61–79
6161:S2CID
6122:(PDF)
6054:[
4725:Then
4639:From
3674:Let
2454:Euler
2185:>
6343:list
6265:and
6203:ISBN
5974:does
5952:even
5927:+ 1)
5905:even
5892:even
5887:even
5846:even
5839:even
5774:even
5757:+ 1)
5746:even
5719:even
5521:>
5440:>
4661:<
4503:log
4483:log
4352:<
4152:<
4021:<
3955:<
3878:>
3739:<
3650:1 +
3599:<
3576:<
2994:and
2356:<
2280:<
1979:} \
1817:and
1656:<
1570:Let
710:log
256:and
81:The
36:log(
6195:MAA
6153:doi
6099:",
5966:any
5946:odd
5938:+ 1
5902:odd
5879:odd
5812:odd
5771:odd
5740:odd
5733:≥ 1
5713:odd
4936:log
4881:log
4875:log
4860:log
4688:log
4682:log
4667:log
4631:).
4585:log
4579:log
4530:lim
4336:log
4312:log
4306:log
4225:exp
4181:exp
3996:log
3679:≥ 2
3431:log
2427:≥ 2
2406:≥ 2
2013:in
1825:is
1793:in
1769:in
1705:in
1583:sum
1543:as
1509:log
1503:log
1331:log
1319:= 1
1291:log
855:log
785:log
729:log
352:log
328:log
322:log
56:log
54:or
48:ln(
6749::
6201:.
6193:.
6159:.
6149:99
6147:.
6130:17
6128:.
6124:.
6083:78
6076:.
6029:.
4983:.
3410:AB
2437:.
1805:=
1742:≤
1729:≤
1589:.
1567:.
1489:11
400:.
264:.
216:17
203:13
190:11
6345:)
6341:(
6255:e
6248:t
6241:v
6227:.
6211:.
6167:.
6155::
6097:p
6031:9
5962:n
5949:/
5936:n
5932:p
5925:n
5923:(
5897:=
5883:+
5872:=
5864:1
5861:+
5858:n
5854:p
5835:+
5830:1
5827:+
5824:n
5820:p
5805:=
5798:1
5795:+
5792:n
5788:p
5784:1
5779:+
5755:n
5753:(
5743:/
5731:n
5726:n
5716:/
5703:2
5700:/
5697:1
5665:k
5645:1
5637:k
5633:x
5603:=
5598:j
5595:1
5583:1
5580:=
5577:j
5564:k
5560:p
5551:2
5547:p
5541:1
5537:p
5533:+
5530:1
5526:1
5515:)
5510:k
5506:p
5497:2
5493:p
5487:1
5483:p
5479:(
5476:j
5473:+
5470:1
5466:1
5454:1
5451:=
5448:j
5435:i
5431:)
5425:k
5421:x
5417:(
5407:0
5404:=
5401:i
5374:)
5369:k
5365:p
5356:2
5352:p
5346:1
5342:p
5338:(
5335:j
5332:+
5329:1
5309:k
5303:n
5300:,
5295:n
5291:p
5268:i
5264:)
5258:k
5254:x
5250:(
5240:0
5237:=
5234:i
5209:}
5203:,
5198:2
5195:+
5192:k
5188:p
5184:,
5179:1
5176:+
5173:k
5169:p
5165:{
5145:i
5123:i
5119:)
5113:k
5109:x
5105:(
5085:0
5079:i
5057:.
5050:n
5046:p
5042:1
5030:1
5027:+
5024:k
5021:=
5018:n
5010:=
5005:k
5001:x
4988:k
4948:=
4942:n
4933:n
4930:2
4926:1
4914:6
4911:=
4908:n
4887:n
4872:n
4869:+
4866:n
4857:n
4853:1
4841:6
4838:=
4835:n
4813:n
4809:p
4805:1
4793:6
4790:=
4787:n
4768:n
4764:p
4760:1
4748:1
4745:=
4742:n
4711:6
4705:n
4694:n
4679:n
4676:+
4673:n
4664:n
4656:n
4652:p
4619:M
4602:M
4599:=
4595:)
4591:n
4571:p
4568:1
4561:n
4555:p
4546:(
4534:n
4514:6
4511:/
4508:π
4494:3
4491:/
4488:5
4461:6
4456:2
4446:=
4439:2
4435:k
4431:1
4419:1
4416:=
4413:k
4376:p
4373:1
4366:n
4360:p
4347:3
4344:5
4330:)
4327:1
4324:+
4321:n
4318:(
4293:3
4290:/
4287:5
4263:)
4257:p
4254:1
4247:n
4241:p
4232:(
4220:3
4217:5
4212:=
4201:)
4196:p
4193:1
4188:(
4176:n
4170:p
4160:3
4157:5
4138:2
4134:k
4130:1
4123:n
4118:1
4115:=
4112:k
4103:)
4097:p
4094:1
4089:+
4086:1
4082:(
4076:n
4070:p
4050:i
4047:1
4040:n
4035:1
4032:=
4029:i
4014:)
4011:1
4008:+
4005:n
4002:(
3963:3
3960:5
3947:2
3944:1
3939:+
3936:n
3932:1
3922:3
3919:2
3914:+
3911:1
3908:=
3892:2
3888:k
3884:1
3869:4
3866:1
3856:2
3852:k
3847:1
3841:=
3831:)
3822:2
3819:1
3814:+
3811:k
3807:1
3794:2
3791:1
3783:k
3779:1
3773:(
3764:n
3759:2
3756:=
3753:k
3745:+
3742:1
3728:2
3724:k
3720:1
3713:n
3708:1
3705:=
3702:k
3677:n
3671:.
3667:x
3658:)
3656:x
3652:x
3628:i
3625:1
3618:n
3613:1
3610:=
3607:i
3585:i
3582:1
3561:x
3557:x
3554:d
3546:1
3543:+
3540:i
3535:i
3521:n
3516:1
3513:=
3510:i
3502:=
3490:x
3486:x
3483:d
3475:1
3472:+
3469:n
3464:1
3456:=
3449:)
3446:1
3443:+
3440:n
3437:(
3396:i
3392:/
3388:1
3378:B
3362:2
3358:b
3353:/
3349:1
3339:A
3325:)
3320:s
3316:p
3307:2
3303:p
3297:1
3293:p
3289:(
3285:/
3281:1
3257:.
3251:+
3243:s
3239:p
3230:2
3226:p
3220:1
3216:p
3211:1
3206:=
3193:+
3189:)
3182:s
3178:p
3174:1
3169:(
3161:)
3154:2
3150:p
3146:1
3141:(
3136:)
3129:1
3125:p
3121:1
3116:(
3112:=
3104:)
3096:s
3092:p
3088:1
3083:+
3080:1
3076:(
3068:)
3060:2
3056:p
3052:1
3047:+
3044:1
3040:(
3035:)
3027:1
3023:p
3019:1
3014:+
3011:1
3007:(
2982:,
2975:2
2971:b
2967:1
2954:s
2950:p
2941:2
2937:p
2931:1
2927:p
2922:1
2917:=
2912:i
2909:1
2884:.
2881:B
2875:A
2872:=
2868:)
2860:2
2856:k
2852:1
2845:n
2840:1
2837:=
2834:k
2825:(
2817:)
2812:)
2806:p
2803:1
2798:+
2795:1
2791:(
2785:n
2779:p
2770:(
2761:i
2758:1
2751:n
2746:1
2743:=
2740:i
2725:i
2711:,
2706:2
2702:b
2695:)
2690:s
2686:p
2677:2
2673:p
2667:1
2663:p
2659:(
2656:=
2653:i
2643:q
2639:q
2635:β
2621:,
2614:r
2604:+
2599:r
2589:2
2584:r
2580:q
2569:2
2559:+
2554:2
2544:2
2539:2
2535:q
2524:1
2514:+
2509:1
2499:2
2494:1
2490:q
2486:=
2483:i
2469:i
2458:p
2449:n
2432:x
2422:2
2419:/
2415:x
2404:x
2386:)
2383:3
2380:(
2375:|
2369:x
2365:M
2360:|
2351:2
2348:x
2320:i
2316:p
2312:x
2300:1
2297:+
2294:k
2291:=
2288:i
2276:|
2270:x
2267:,
2264:i
2260:N
2255:|
2244:1
2241:+
2238:k
2235:=
2232:i
2220:|
2214:x
2210:M
2205:|
2198:x
2187:x
2182:i
2180:p
2169:i
2167:p
2163:/
2159:x
2148:x
2146:,
2144:i
2140:N
2121:x
2118:,
2115:i
2111:N
2100:1
2097:+
2094:k
2091:=
2088:i
2080:=
2075:x
2071:M
2064:}
2061:x
2058:,
2052:,
2049:2
2046:,
2043:1
2040:{
2029:i
2027:p
2023:i
2019:}
2017:x
2011:n
2005:x
2003:,
2001:i
1997:N
1991:k
1989:p
1983:x
1981:M
1977:x
1964:x
1962:M
1957:x
1935:)
1932:2
1929:(
1923:x
1916:k
1912:2
1904:|
1898:x
1894:M
1889:|
1878:m
1871:x
1863:r
1859:2
1855:r
1845:k
1843:p
1839:1
1836:p
1831:k
1823:r
1819:r
1815:m
1810:r
1807:m
1803:n
1797:x
1795:M
1791:n
1778:x
1773:x
1771:M
1759:x
1757:M
1748:k
1744:p
1739:i
1737:p
1731:x
1727:n
1721:k
1719:p
1711:}
1709:x
1703:n
1698:x
1696:M
1692:x
1676:)
1673:1
1670:(
1664:2
1661:1
1649:i
1645:p
1641:1
1629:1
1626:+
1623:k
1620:=
1617:i
1600:k
1579:i
1574:i
1572:p
1545:n
1540:n
1534:n
1518:.
1500:=
1494:+
1486:1
1481:+
1476:7
1473:1
1468:+
1463:5
1460:1
1455:+
1450:3
1447:1
1442:+
1437:2
1434:1
1429:=
1426:A
1404:.
1399:n
1394:n
1390:x
1377:1
1374:=
1371:n
1363:=
1359:)
1353:x
1347:1
1343:1
1338:(
1317:x
1300:,
1288:=
1283:n
1280:1
1268:1
1265:=
1262:n
1243:K
1222:K
1219:+
1216:A
1213:=
1200:+
1197:D
1192:4
1189:1
1184:+
1181:C
1176:3
1173:1
1168:+
1165:B
1160:2
1157:1
1152:+
1149:A
1146:=
1133:+
1126:4
1122:p
1118:1
1111:p
1101:4
1098:1
1093:+
1086:3
1082:p
1078:1
1071:p
1061:3
1058:1
1053:+
1046:2
1042:p
1038:1
1031:p
1021:2
1018:1
1013:+
1008:p
1005:1
998:p
990:=
979:)
972:+
964:3
960:p
956:3
952:1
947:+
939:2
935:p
931:2
927:1
922:+
917:p
914:1
908:(
902:p
894:=
883:)
877:p
874:1
866:1
862:(
850:p
839:=
835:)
826:1
819:p
812:1
808:1
801:p
792:(
782:=
772:)
766:n
763:1
751:1
748:=
745:n
736:(
712:x
661:1
654:p
647:1
643:1
636:p
628:=
624:)
617:+
610:2
606:p
602:1
597:+
592:p
589:1
584:+
581:1
577:(
571:p
563:=
558:n
555:1
543:1
540:=
537:n
501:=
495:+
490:4
487:1
482:+
477:3
474:1
469:+
464:2
461:1
456:+
453:1
450:=
445:n
442:1
430:1
427:=
424:n
392:(
386:n
370:6
365:2
346:)
343:1
340:+
337:n
334:(
314:p
311:1
302:n
296:p
286:p
227:=
221:+
213:1
208:+
200:1
195:+
187:1
182:+
177:7
174:1
169:+
164:5
161:1
156:+
151:3
148:1
143:+
138:2
135:1
130:=
125:p
122:1
110:p
68:.
66:)
64:x
62:(
59:e
52:)
50:x
40:)
38:x
20:)
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