2233:
1663:
2868:
2860:
3204:
2228:{\displaystyle {\begin{aligned}b^{-\delta _{b}(n)}\sum _{k=b^{n-1}}^{b^{n}-1}{\frac {k}{b^{n(k-b^{n-1}+1)}}}&=b^{-\delta _{b}(n)}b^{n(b^{n-1}-1)}\left(\sum _{k=b^{n-1}}^{\infty }{\frac {k}{b^{nk}}}-\sum _{k=b^{n}}^{\infty }{\frac {k}{b^{nk}}}\right)\\&={\frac {b^{2n-1}-b^{n-1}+1}{\left(b^{n}-1\right)^{2}}}b^{-\delta _{b}(n)}-{\frac {b^{2n}-b^{n}+1}{\left(b^{n}-1\right)^{2}}}b^{-\delta _{b}(n+1)}.\end{aligned}}}
3028:
2530:
3213:
The first and second incrementally largest terms ("high-water marks") after the initial zero are 8 and 9, respectively, and occur at positions 1 and 2. Sikora (2012) noticed that the number of digits in the high-water marks starting with the fourth display an apparent pattern. Indeed, the high-water
2923:
The large number at position 18 has 166 digits, and the next very large term at position 40 of the continued fraction has 2504 digits. That there are such large numbers as terms of the continued fraction expansion means that the convergents obtained by stopping before these large numbers
2245:
1214:
3199:{\displaystyle {\begin{aligned}{\frac {60499999499}{490050000000}}&=0.123456789+10^{-9}\sum _{k=10}^{\infty }k/10^{2(k-9)}=0.123456789+10^{-9}{\frac {991}{9801}}\\&=0.123456789{\overline {10111213141516171819\ldots 90919293949596979900010203040506070809}},\end{aligned}}}
3489:
1063:
820:
1658:
2628:. As a consequence, each additional term provides an exponentially growing number of correct digits even though the number of digits in the numerators and denominators of the fractions comprising these terms grows only linearly. For example, the first few terms of
1068:
446:
1434:
3015:
3343:
932:
908:
644:
1477:
3493:
However, it is still unknown as to whether or not there is a way to determine where the large terms (with at least 6 digits) occur, or their values. The high-water marks themselves are located at positions
2525:{\displaystyle C_{b}={\frac {b}{(b-1)^{2}}}-\sum _{n=1}^{\infty }\left({\frac {b^{2n}-b^{n}+1}{\left(b^{n}-1\right)^{2}}}-{\frac {b^{2n+1}-b^{n}+1}{\left(b^{n+1}-1\right)^{2}}}\right)b^{-\delta _{b}(n+1)}.}
3033:
1668:
231:
If we denote a digit string as , then, in base 10, we would expect strings , , , …, to occur 1/10 of the time, strings , , ..., , to occur 1/100 of the time, and so on, in a normal number.
2850:
213:
if its digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. A number
1240:
2620:
2565:
1266:
529:
167:) is any sequence of digits obtained by concatenating all finite digit-strings (in any given base) in some recursive order. For instance, the binary Champernowne sequence in
1472:
1309:
571:
3269:
3021:. Truncating just before the 18th partial quotient gives an approximation that matches the first two terms of the series, that is, the terms up to the term containing
2863:
The first 161 quotients of the continued fraction of the
Champernowne constant. The 4th, 18th, 40th, and 101st are much bigger than 270, so do not appear on the graph.
3017:
which matches the first term in the rapidly converging series expansion of the previous section and which approximates
Champernowne's constant with an error of about
597:
363:
1318:
487:
417:
390:
259:
3239:
2931:
337:
286:
306:
3505:
2916:
422:
185:
152:
77:
3765:
3594:
825:
3853:
1209:{\displaystyle C_{b}=\sum _{n=1}^{\infty }n\cdot b^{-\left(n+\sum \limits _{k=1}^{n-1}\left\lfloor \log _{b}(k+1)\right\rfloor \right)},}
2640:
3484:{\displaystyle d_{n}={\frac {13-67\times 10^{n-3}}{45}}+\left(2^{n}5^{n-3}-2\right),n\in \mathbb {Z} \cap \left[3,\infty \right).}
2895:(because it is not an irreducible quadratic). A simple continued fraction is a continued fraction where the denominator is 1. The
1058:{\displaystyle C_{b}=\sum _{n=1}^{\infty }n\cdot b^{-\left(\sum \limits _{k=1}^{n}\left\lceil \log _{b}(k+1)\right\rceil \right)}}
2899:
expansion of
Champernowne's constant exhibits extremely large terms appearing between many small ones. For example, in base 10,
815:{\displaystyle C_{10}=\sum _{n=1}^{\infty }10^{-\delta _{10}(n)}\sum _{k=10^{n-1}}^{10^{n}-1}{\frac {k}{10^{n(k-10^{n-1}+1)}}},}
3843:
3518:
3533:
1653:{\displaystyle \delta _{b}(n)=(b-1)\sum _{\ell =1}^{n-1}b^{\ell -1}\ell ={\frac {1}{b-1}}\left(1+b^{n-1}((b-1)n-b)\right),}
3858:
3788:
1269:
615:
3757:
3586:
2925:
623:
3686:
261:
is normal in base 10, while Nakai and
Shiokawa proved a more general theorem, a corollary of which is that
3848:
3634:
3340:
whose pattern becomes obvious starting with the 6th high-water mark. The number of terms can be given by
619:
459:
455:
218:
42:
36:
3736:
Sikora, J. K. "On the High Water Mark
Convergents of Champernowne's Constant in Base Ten." 3 Oct 2012.
1219:
192:
where spaces (otherwise to be ignored) have been inserted just to show the strings being concatenated.
2583:
2535:
3749:
3681:
1245:
603:
492:
1441:
1278:
3639:
2928:
of the
Champernowne constant. For example, truncating just before the 4th partial quotient, gives
534:
419:
is 0.123456789101112131415161718192021222324252627282930313. When we express this in base 9 we get
3676:
3248:
2867:
3783:
3622:
3618:
2896:
2892:
2884:
2880:
1275:
Returning to the first of these series, both the summand of the outer sum and the expression for
46:
45:
whose decimal expansion has important properties. It is named after economist and mathematician
3761:
3713:
3590:
2872:
576:
342:
3820:
3811:
3797:
3771:
3600:
3527:
1312:
465:
395:
368:
237:
3217:
315:
264:
3775:
3716:
3604:
3578:
2888:
638:
3809:
Nakai, Y.; Shiokawa, I. (1992), "Discrepancy estimates for a class of normal numbers",
3700:
Analysis of the High Water Mark
Convergents of Champernowne's Constant in Various Bases
3626:
291:
168:
2859:
84:
Champernowne constants can also be constructed in other bases similarly; for example,
3837:
910:
is the number of digits between the decimal point and the first contribution from an
209:
50:
2871:
The first 161 quotients of the continued fraction of the
Champernowne constant on a
3522:
3536:, another number obtained through concatenation a representation in a given base.
451:
441:{\displaystyle {0.10888888853823026326512111305027757201400001517660835887}_{9}}
201:
39:
20:
1429:{\displaystyle \sum _{k=n}^{\infty }ka^{k}=a^{n}{\frac {n-(n-1)a}{(1-a)^{2}}}.}
3801:
3756:. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge:
3682:
Approximation to certain transcendental decimal fractions by algebraic numbers
3010:{\displaystyle 10/81=\sum _{k=1}^{\infty }k/10^{k}=0.{\overline {123456790}},}
914:-digit base-10 number; these expressions generalize to an arbitrary base
3825:
3721:
2532:
Observe that in the summand, the expression in parentheses is approximately
627:
49:, who published it as an undergraduate in 1933. The number is defined by
637:
The definition of the
Champernowne constant immediately gives rise to an
611:
903:{\displaystyle \delta _{10}(n)=9\sum _{\ell =1}^{n-1}10^{\ell -1}\ell }
54:
3214:
marks themselves grow doubly-exponentially, and the number of digits
3667:, Proc. Konin. Neder. Akad. Wet. Ser. A. 40 (1937), p. 421–428.
3498:
1, 2, 4, 18, 40, 162, 526, 1708, 4838, 13522, 34062, .... (sequence
3206:
which approximates
Champernowne's constant with error approximately
3786:(1933), "The construction of decimals normal in the scale of ten",
3699:
2866:
2858:
3737:
138:
obtained by writing it in base 10 and juxtaposing the digits:
16:
Transcendental number(s) with all positive integers in order
3748:
Cassaigne, J.; Nicolas, F. (2010). "Factor complexity". In
3665:
Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen
3500:
2911:
180:
147:
72:
3583:
Automatic Sequences: Theory, Applications, Generalizations
429:
0.10888888853823026326512111305027757201400001517660835887
3530:, another constant defined by its decimal representation
3633:, University of Auckland, New Zealand, pp. 1–35,
3346:
3251:
3220:
3031:
2934:
2643:
2586:
2538:
2248:
1666:
1480:
1444:
1321:
1281:
1248:
1222:
1071:
935:
828:
647:
579:
537:
495:
468:
425:
398:
371:
345:
318:
294:
267:
240:
3702:, in: arXiv:1408.0261, 1 Aug 2014, see Definition 9
3689:, Volume 37, Issue 2, February 1991, Pages 231–241
3483:
3263:
3233:
3198:
3009:
2844:
2614:
2559:
2524:
2227:
1652:
1466:
1428:
1303:
1260:
1234:
1208:
1057:
902:
814:
591:
565:
523:
481:
440:
411:
384:
357:
331:
300:
280:
253:
1311:can be simplified using the closed form for the
392:is normal in base 9. For example, 54 digits of
2883:expansion of Champernowne's constant does not
2845:{\displaystyle C_{10}={\frac {10}{81}}-\left.}
3652:
3521:, a similar normal number, defined using the
8:
3556:
1255:
1249:
1229:
1223:
1660:while the summand of the outer sum becomes
3789:Journal of the London Mathematical Society
3754:Combinatorics, automata, and number theory
57:representations of the positive integers:
3824:
3638:
3452:
3451:
3419:
3409:
3379:
3360:
3351:
3345:
3250:
3225:
3219:
3171:
3148:
3139:
3105:
3096:
3087:
3076:
3063:
3036:
3032:
3030:
2994:
2982:
2973:
2964:
2953:
2938:
2933:
2819:
2797:
2784:
2767:
2745:
2732:
2715:
2693:
2680:
2657:
2648:
2642:
2591:
2585:
2539:
2537:
2496:
2488:
2471:
2448:
2425:
2403:
2396:
2385:
2368:
2345:
2329:
2322:
2311:
2300:
2284:
2262:
2253:
2247:
2195:
2187:
2175:
2158:
2135:
2119:
2112:
2092:
2084:
2072:
2055:
2026:
2004:
1997:
1971:
1962:
1956:
1949:
1938:
1920:
1911:
1905:
1892:
1881:
1849:
1838:
1817:
1809:
1773:
1756:
1747:
1733:
1728:
1715:
1704:
1683:
1675:
1667:
1665:
1600:
1567:
1549:
1533:
1522:
1485:
1479:
1449:
1443:
1414:
1369:
1363:
1350:
1337:
1326:
1320:
1286:
1280:
1247:
1221:
1167:
1146:
1135:
1116:
1100:
1089:
1076:
1070:
1019:
1004:
993:
980:
964:
953:
940:
934:
885:
869:
858:
833:
827:
784:
767:
758:
744:
739:
726:
715:
694:
686:
676:
665:
652:
646:
578:
548:
536:
506:
494:
473:
467:
432:
427:
424:
403:
397:
376:
370:
344:
323:
317:
293:
272:
266:
245:
239:
3552:
3550:
3546:
641:representation involving a double sum,
3181:90919293949596979900010203040506070809
2574:and rapidly approaches that value as
7:
3567:Cassaigne & Nicolas (2010) p.165
929:respectively. Alternative forms are
1132:
990:
3631:Disjunctive sequences: An overview
3470:
3088:
2965:
2312:
1957:
1906:
1338:
1101:
965:
677:
365:. For example, it is not known if
14:
1235:{\displaystyle \lfloor x\rfloor }
2615:{\displaystyle \delta _{b}(n+1)}
2560:{\displaystyle {\frac {b-1}{b}}}
1313:two-dimensional geometric series
312:. It is an open problem whether
1261:{\displaystyle \lceil x\rceil }
524:{\displaystyle \mu (C_{10})=10}
228:follow a uniform distribution.
68:= 0.12345678910111213141516...
3738:http://arxiv.org/abs/1210.1263
3121:
3109:
2924:provide an exceptionally good
2609:
2597:
2514:
2502:
2281:
2268:
2213:
2201:
2104:
2098:
1867:
1842:
1829:
1823:
1791:
1760:
1695:
1689:
1639:
1627:
1615:
1612:
1515:
1503:
1497:
1491:
1467:{\displaystyle \delta _{b}(n)}
1461:
1455:
1411:
1398:
1390:
1378:
1304:{\displaystyle \delta _{b}(n)}
1298:
1292:
1188:
1176:
1040:
1028:
845:
839:
802:
771:
706:
700:
554:
541:
512:
499:
1:
2887:(because the constant is not
1438:The resulting expression for
566:{\displaystyle \mu (C_{b})=b}
131:is the sequence of digits of
3264:{\displaystyle n\geqslant 3}
3184:
2999:
2855:Continued fraction expansion
454:showed that the constant is
3854:Real transcendental numbers
1270:floor and ceiling functions
918:by replacing 10 and 9 with
602:The Champernowne word is a
176:0 1 00 01 10 11 000 001 ...
143:12345678910111213141516...
3875:
3758:Cambridge University Press
3587:Cambridge University Press
2580:grows, while the exponent
3653:Nakai & Shiokawa 1992
3534:Smarandache–Wellin number
2897:simple continued fraction
2881:simple continued fraction
2622:grows exponentially with
234:Champernowne proved that
163:(sometimes also called a
95:= 0.11011100101110111...
3687:Journal of Number Theory
217:is said to be normal in
3826:10.4064/aa-62-3-271-284
3802:10.1112/jlms/s1-8.4.254
3752:; Rigo, Michel (eds.).
3717:"Champernowne constant"
3519:Copeland–Erdős constant
592:{\displaystyle b\geq 2}
358:{\displaystyle b\neq k}
3844:Mathematical constants
3485:
3265:
3235:
3200:
3092:
3011:
2969:
2876:
2864:
2846:
2616:
2561:
2526:
2316:
2229:
1961:
1910:
1746:
1654:
1544:
1468:
1430:
1342:
1305:
1262:
1236:
1210:
1157:
1105:
1059:
1009:
969:
904:
880:
816:
757:
681:
593:
567:
525:
483:
482:{\displaystyle C_{10}}
442:
413:
412:{\displaystyle C_{10}}
386:
385:{\displaystyle C_{10}}
359:
333:
302:
282:
255:
254:{\displaystyle C_{10}}
224:if its digits in base
114:= 0.12101112202122...
3577:Allouche, Jean-Paul;
3486:
3266:
3236:
3234:{\displaystyle d_{n}}
3201:
3072:
3012:
2949:
2870:
2862:
2847:
2617:
2562:
2527:
2296:
2230:
1934:
1877:
1700:
1655:
1518:
1469:
1431:
1322:
1306:
1263:
1237:
1211:
1131:
1085:
1060:
989:
949:
905:
854:
817:
711:
661:
594:
568:
531:, and more generally
526:
484:
460:irrationality measure
443:
414:
387:
360:
334:
332:{\displaystyle C_{k}}
303:
283:
281:{\displaystyle C_{b}}
256:
161:Champernowne sequence
25:Champernowne constant
3859:Sequences and series
3760:. pp. 163–247.
3528:Liouville's constant
3344:
3249:
3218:
3175:10111213141516171819
3029:
2932:
2641:
2584:
2536:
2246:
1664:
1478:
1442:
1319:
1279:
1246:
1220:
1069:
933:
826:
645:
608:disjunctive sequence
604:disjunctive sequence
577:
535:
493:
466:
423:
396:
369:
343:
316:
292:
265:
238:
3784:Champernowne, D. G.
339:is normal in bases
3714:Weisstein, Eric W.
3481:
3261:
3231:
3196:
3194:
3007:
2877:
2865:
2842:
2612:
2557:
2522:
2225:
2223:
1650:
1464:
1426:
1301:
1258:
1232:
1206:
1055:
900:
812:
589:
563:
521:
479:
438:
409:
382:
355:
329:
298:
288:is normal in base
278:
251:
159:More generally, a
47:D. G. Champernowne
3767:978-0-521-51597-9
3596:978-0-521-82332-6
3557:Champernowne 1933
3395:
3187:
3156:
3044:
3002:
2873:logarithmic scale
2805:
2792:
2753:
2740:
2701:
2688:
2665:
2555:
2477:
2391:
2291:
2235:Summing over all
2181:
2078:
1980:
1929:
1796:
1583:
1421:
807:
622:) in which every
301:{\displaystyle b}
165:Champernowne word
125:Champernowne word
3866:
3829:
3828:
3812:Acta Arithmetica
3804:
3779:
3740:
3734:
3728:
3727:
3726:
3709:
3703:
3698:John K. Sikora:
3696:
3690:
3674:
3668:
3663:K. Mahler,
3661:
3655:
3650:
3644:
3643:
3642:
3615:
3609:
3608:
3579:Shallit, Jeffrey
3574:
3568:
3565:
3559:
3554:
3503:
3490:
3488:
3487:
3482:
3477:
3473:
3455:
3441:
3437:
3430:
3429:
3414:
3413:
3396:
3391:
3390:
3389:
3361:
3356:
3355:
3335:
3332:
3329:
3325:
3322:
3319:
3315:
3312:
3309:
3305:
3302:
3299:
3295:
3292:
3289:
3285:
3282:
3278:
3270:
3268:
3267:
3262:
3240:
3238:
3237:
3232:
3230:
3229:
3209:
3205:
3203:
3202:
3197:
3195:
3188:
3183:
3172:
3161:
3157:
3149:
3147:
3146:
3125:
3124:
3100:
3091:
3086:
3071:
3070:
3045:
3037:
3024:
3020:
3016:
3014:
3013:
3008:
3003:
2995:
2987:
2986:
2977:
2968:
2963:
2942:
2914:
2851:
2849:
2848:
2843:
2838:
2834:
2827:
2826:
2811:
2807:
2806:
2798:
2793:
2785:
2775:
2774:
2759:
2755:
2754:
2746:
2741:
2733:
2723:
2722:
2707:
2703:
2702:
2694:
2689:
2681:
2666:
2658:
2653:
2652:
2636:
2627:
2621:
2619:
2618:
2613:
2596:
2595:
2579:
2573:
2566:
2564:
2563:
2558:
2556:
2551:
2540:
2531:
2529:
2528:
2523:
2518:
2517:
2501:
2500:
2483:
2479:
2478:
2476:
2475:
2470:
2466:
2459:
2458:
2437:
2430:
2429:
2417:
2416:
2397:
2392:
2390:
2389:
2384:
2380:
2373:
2372:
2357:
2350:
2349:
2337:
2336:
2323:
2315:
2310:
2292:
2290:
2289:
2288:
2263:
2258:
2257:
2241:
2234:
2232:
2231:
2226:
2224:
2217:
2216:
2200:
2199:
2182:
2180:
2179:
2174:
2170:
2163:
2162:
2147:
2140:
2139:
2127:
2126:
2113:
2108:
2107:
2097:
2096:
2079:
2077:
2076:
2071:
2067:
2060:
2059:
2044:
2037:
2036:
2018:
2017:
1998:
1990:
1986:
1982:
1981:
1979:
1978:
1963:
1960:
1955:
1954:
1953:
1930:
1928:
1927:
1912:
1909:
1904:
1903:
1902:
1871:
1870:
1860:
1859:
1833:
1832:
1822:
1821:
1797:
1795:
1794:
1784:
1783:
1748:
1745:
1738:
1737:
1727:
1726:
1725:
1699:
1698:
1688:
1687:
1659:
1657:
1656:
1651:
1646:
1642:
1611:
1610:
1584:
1582:
1568:
1560:
1559:
1543:
1532:
1490:
1489:
1473:
1471:
1470:
1465:
1454:
1453:
1435:
1433:
1432:
1427:
1422:
1420:
1419:
1418:
1396:
1370:
1368:
1367:
1355:
1354:
1341:
1336:
1310:
1308:
1307:
1302:
1291:
1290:
1267:
1265:
1264:
1259:
1241:
1239:
1238:
1233:
1215:
1213:
1212:
1207:
1202:
1201:
1200:
1196:
1195:
1191:
1172:
1171:
1156:
1145:
1104:
1099:
1081:
1080:
1064:
1062:
1061:
1056:
1054:
1053:
1052:
1048:
1047:
1043:
1024:
1023:
1008:
1003:
968:
963:
945:
944:
928:
921:
917:
913:
909:
907:
906:
901:
896:
895:
879:
868:
838:
837:
821:
819:
818:
813:
808:
806:
805:
795:
794:
759:
756:
749:
748:
738:
737:
736:
710:
709:
699:
698:
680:
675:
657:
656:
598:
596:
595:
590:
572:
570:
569:
564:
553:
552:
530:
528:
527:
522:
511:
510:
488:
486:
485:
480:
478:
477:
447:
445:
444:
439:
437:
436:
431:
418:
416:
415:
410:
408:
407:
391:
389:
388:
383:
381:
380:
364:
362:
361:
356:
338:
336:
335:
330:
328:
327:
307:
305:
304:
299:
287:
285:
284:
279:
277:
276:
260:
258:
257:
252:
250:
249:
183:
177:
150:
144:
118:
99:
75:
69:
34:
3874:
3873:
3869:
3868:
3867:
3865:
3864:
3863:
3834:
3833:
3808:
3782:
3768:
3750:Berthé, Valérie
3747:
3744:
3743:
3735:
3731:
3712:
3711:
3710:
3706:
3697:
3693:
3675:
3671:
3662:
3658:
3651:
3647:
3617:
3616:
3612:
3597:
3589:. p. 299.
3576:
3575:
3571:
3566:
3562:
3555:
3548:
3543:
3515:
3499:
3463:
3459:
3415:
3405:
3404:
3400:
3375:
3362:
3347:
3342:
3341:
3333:
3330:
3327:
3323:
3320:
3317:
3313:
3310:
3307:
3303:
3300:
3297:
3293:
3290:
3287:
3283:
3280:
3276:
3247:
3246:
3221:
3216:
3215:
3207:
3193:
3192:
3173:
3159:
3158:
3135:
3101:
3059:
3046:
3027:
3026:
3022:
3018:
2978:
2930:
2929:
2910:
2908:
2857:
2815:
2783:
2779:
2763:
2731:
2727:
2711:
2679:
2675:
2674:
2670:
2644:
2639:
2638:
2635:
2629:
2623:
2587:
2582:
2581:
2575:
2568:
2541:
2534:
2533:
2492:
2484:
2444:
2443:
2439:
2438:
2421:
2399:
2398:
2364:
2363:
2359:
2358:
2341:
2325:
2324:
2321:
2317:
2280:
2267:
2249:
2244:
2243:
2236:
2222:
2221:
2191:
2183:
2154:
2153:
2149:
2148:
2131:
2115:
2114:
2088:
2080:
2051:
2050:
2046:
2045:
2022:
2000:
1999:
1988:
1987:
1967:
1945:
1916:
1888:
1876:
1872:
1845:
1834:
1813:
1805:
1798:
1769:
1752:
1729:
1711:
1679:
1671:
1662:
1661:
1596:
1589:
1585:
1572:
1545:
1481:
1476:
1475:
1445:
1440:
1439:
1410:
1397:
1371:
1359:
1346:
1317:
1316:
1282:
1277:
1276:
1244:
1243:
1218:
1217:
1163:
1162:
1158:
1124:
1120:
1112:
1072:
1067:
1066:
1015:
1014:
1010:
988:
984:
976:
936:
931:
930:
923:
919:
915:
911:
881:
829:
824:
823:
780:
763:
740:
722:
690:
682:
648:
643:
642:
639:infinite series
635:
614:(over a finite
610:is an infinite
575:
574:
544:
533:
532:
502:
491:
490:
469:
464:
463:
426:
421:
420:
399:
394:
393:
372:
367:
366:
341:
340:
319:
314:
313:
290:
289:
268:
263:
262:
241:
236:
235:
198:
179:
175:
146:
142:
137:
117:
113:
107:
98:
94:
88:
71:
67:
61:
33:
27:
17:
12:
11:
5:
3872:
3870:
3862:
3861:
3856:
3851:
3846:
3836:
3835:
3832:
3831:
3819:(3): 271–284,
3806:
3796:(4): 254–260,
3780:
3766:
3742:
3741:
3729:
3704:
3691:
3669:
3656:
3645:
3640:10.1.1.34.1370
3610:
3595:
3569:
3560:
3545:
3544:
3542:
3539:
3538:
3537:
3531:
3525:
3514:
3511:
3510:
3509:
3480:
3476:
3472:
3469:
3466:
3462:
3458:
3454:
3450:
3447:
3444:
3440:
3436:
3433:
3428:
3425:
3422:
3418:
3412:
3408:
3403:
3399:
3394:
3388:
3385:
3382:
3378:
3374:
3371:
3368:
3365:
3359:
3354:
3350:
3338:
3337:
3260:
3257:
3254:
3228:
3224:
3191:
3186:
3182:
3179:
3176:
3170:
3167:
3164:
3162:
3160:
3155:
3152:
3145:
3142:
3138:
3134:
3131:
3128:
3123:
3120:
3117:
3114:
3111:
3108:
3104:
3099:
3095:
3090:
3085:
3082:
3079:
3075:
3069:
3066:
3062:
3058:
3055:
3052:
3049:
3047:
3043:
3040:
3035:
3034:
3006:
3001:
2998:
2993:
2990:
2985:
2981:
2976:
2972:
2967:
2962:
2959:
2956:
2952:
2948:
2945:
2941:
2937:
2921:
2920:
2909:= . (sequence
2906:
2856:
2853:
2841:
2837:
2833:
2830:
2825:
2822:
2818:
2814:
2810:
2804:
2801:
2796:
2791:
2788:
2782:
2778:
2773:
2770:
2766:
2762:
2758:
2752:
2749:
2744:
2739:
2736:
2730:
2726:
2721:
2718:
2714:
2710:
2706:
2700:
2697:
2692:
2687:
2684:
2678:
2673:
2669:
2664:
2661:
2656:
2651:
2647:
2633:
2611:
2608:
2605:
2602:
2599:
2594:
2590:
2554:
2550:
2547:
2544:
2521:
2516:
2513:
2510:
2507:
2504:
2499:
2495:
2491:
2487:
2482:
2474:
2469:
2465:
2462:
2457:
2454:
2451:
2447:
2442:
2436:
2433:
2428:
2424:
2420:
2415:
2412:
2409:
2406:
2402:
2395:
2388:
2383:
2379:
2376:
2371:
2367:
2362:
2356:
2353:
2348:
2344:
2340:
2335:
2332:
2328:
2320:
2314:
2309:
2306:
2303:
2299:
2295:
2287:
2283:
2279:
2276:
2273:
2270:
2266:
2261:
2256:
2252:
2220:
2215:
2212:
2209:
2206:
2203:
2198:
2194:
2190:
2186:
2178:
2173:
2169:
2166:
2161:
2157:
2152:
2146:
2143:
2138:
2134:
2130:
2125:
2122:
2118:
2111:
2106:
2103:
2100:
2095:
2091:
2087:
2083:
2075:
2070:
2066:
2063:
2058:
2054:
2049:
2043:
2040:
2035:
2032:
2029:
2025:
2021:
2016:
2013:
2010:
2007:
2003:
1996:
1993:
1991:
1989:
1985:
1977:
1974:
1970:
1966:
1959:
1952:
1948:
1944:
1941:
1937:
1933:
1926:
1923:
1919:
1915:
1908:
1901:
1898:
1895:
1891:
1887:
1884:
1880:
1875:
1869:
1866:
1863:
1858:
1855:
1852:
1848:
1844:
1841:
1837:
1831:
1828:
1825:
1820:
1816:
1812:
1808:
1804:
1801:
1799:
1793:
1790:
1787:
1782:
1779:
1776:
1772:
1768:
1765:
1762:
1759:
1755:
1751:
1744:
1741:
1736:
1732:
1724:
1721:
1718:
1714:
1710:
1707:
1703:
1697:
1694:
1691:
1686:
1682:
1678:
1674:
1670:
1669:
1649:
1645:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1609:
1606:
1603:
1599:
1595:
1592:
1588:
1581:
1578:
1575:
1571:
1566:
1563:
1558:
1555:
1552:
1548:
1542:
1539:
1536:
1531:
1528:
1525:
1521:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1488:
1484:
1463:
1460:
1457:
1452:
1448:
1425:
1417:
1413:
1409:
1406:
1403:
1400:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1366:
1362:
1358:
1353:
1349:
1345:
1340:
1335:
1332:
1329:
1325:
1300:
1297:
1294:
1289:
1285:
1257:
1254:
1251:
1231:
1228:
1225:
1205:
1199:
1194:
1190:
1187:
1184:
1181:
1178:
1175:
1170:
1166:
1161:
1155:
1152:
1149:
1144:
1141:
1138:
1134:
1130:
1127:
1123:
1119:
1115:
1111:
1108:
1103:
1098:
1095:
1092:
1088:
1084:
1079:
1075:
1051:
1046:
1042:
1039:
1036:
1033:
1030:
1027:
1022:
1018:
1013:
1007:
1002:
999:
996:
992:
987:
983:
979:
975:
972:
967:
962:
959:
956:
952:
948:
943:
939:
899:
894:
891:
888:
884:
878:
875:
872:
867:
864:
861:
857:
853:
850:
847:
844:
841:
836:
832:
811:
804:
801:
798:
793:
790:
787:
783:
779:
776:
773:
770:
766:
762:
755:
752:
747:
743:
735:
732:
729:
725:
721:
718:
714:
708:
705:
702:
697:
693:
689:
685:
679:
674:
671:
668:
664:
660:
655:
651:
634:
631:
588:
585:
582:
562:
559:
556:
551:
547:
543:
540:
520:
517:
514:
509:
505:
501:
498:
476:
472:
456:transcendental
435:
430:
406:
402:
379:
375:
354:
351:
348:
326:
322:
297:
275:
271:
248:
244:
207:is said to be
197:
194:
190:
189:
169:shortlex order
157:
156:
135:
121:
120:
115:
111:
101:
100:
96:
92:
82:
81:
65:
37:transcendental
31:
15:
13:
10:
9:
6:
4:
3:
2:
3871:
3860:
3857:
3855:
3852:
3850:
3849:Number theory
3847:
3845:
3842:
3841:
3839:
3827:
3822:
3818:
3814:
3813:
3807:
3803:
3799:
3795:
3791:
3790:
3785:
3781:
3777:
3773:
3769:
3763:
3759:
3755:
3751:
3746:
3745:
3739:
3733:
3730:
3724:
3723:
3718:
3715:
3708:
3705:
3701:
3695:
3692:
3688:
3684:
3683:
3678:
3673:
3670:
3666:
3660:
3657:
3654:
3649:
3646:
3641:
3636:
3632:
3628:
3624:
3620:
3614:
3611:
3606:
3602:
3598:
3592:
3588:
3584:
3580:
3573:
3570:
3564:
3561:
3558:
3553:
3551:
3547:
3540:
3535:
3532:
3529:
3526:
3524:
3523:prime numbers
3520:
3517:
3516:
3512:
3507:
3502:
3497:
3496:
3495:
3491:
3478:
3474:
3467:
3464:
3460:
3456:
3448:
3445:
3442:
3438:
3434:
3431:
3426:
3423:
3420:
3416:
3410:
3406:
3401:
3397:
3392:
3386:
3383:
3380:
3376:
3372:
3369:
3366:
3363:
3357:
3352:
3348:
3274:
3273:
3272:
3258:
3255:
3252:
3244:
3226:
3222:
3211:
3189:
3180:
3177:
3174:
3168:
3165:
3163:
3153:
3150:
3143:
3140:
3136:
3132:
3129:
3126:
3118:
3115:
3112:
3106:
3102:
3097:
3093:
3083:
3080:
3077:
3073:
3067:
3064:
3060:
3056:
3053:
3050:
3048:
3041:
3038:
3004:
2996:
2991:
2988:
2983:
2979:
2974:
2970:
2960:
2957:
2954:
2950:
2946:
2943:
2939:
2935:
2927:
2926:approximation
2918:
2913:
2905:
2902:
2901:
2900:
2898:
2894:
2890:
2886:
2882:
2874:
2869:
2861:
2854:
2852:
2839:
2835:
2831:
2828:
2823:
2820:
2816:
2812:
2808:
2802:
2799:
2794:
2789:
2786:
2780:
2776:
2771:
2768:
2764:
2760:
2756:
2750:
2747:
2742:
2737:
2734:
2728:
2724:
2719:
2716:
2712:
2708:
2704:
2698:
2695:
2690:
2685:
2682:
2676:
2671:
2667:
2662:
2659:
2654:
2649:
2645:
2632:
2626:
2606:
2603:
2600:
2592:
2588:
2578:
2571:
2552:
2548:
2545:
2542:
2519:
2511:
2508:
2505:
2497:
2493:
2489:
2485:
2480:
2472:
2467:
2463:
2460:
2455:
2452:
2449:
2445:
2440:
2434:
2431:
2426:
2422:
2418:
2413:
2410:
2407:
2404:
2400:
2393:
2386:
2381:
2377:
2374:
2369:
2365:
2360:
2354:
2351:
2346:
2342:
2338:
2333:
2330:
2326:
2318:
2307:
2304:
2301:
2297:
2293:
2285:
2277:
2274:
2271:
2264:
2259:
2254:
2250:
2239:
2218:
2210:
2207:
2204:
2196:
2192:
2188:
2184:
2176:
2171:
2167:
2164:
2159:
2155:
2150:
2144:
2141:
2136:
2132:
2128:
2123:
2120:
2116:
2109:
2101:
2093:
2089:
2085:
2081:
2073:
2068:
2064:
2061:
2056:
2052:
2047:
2041:
2038:
2033:
2030:
2027:
2023:
2019:
2014:
2011:
2008:
2005:
2001:
1994:
1992:
1983:
1975:
1972:
1968:
1964:
1950:
1946:
1942:
1939:
1935:
1931:
1924:
1921:
1917:
1913:
1899:
1896:
1893:
1889:
1885:
1882:
1878:
1873:
1864:
1861:
1856:
1853:
1850:
1846:
1839:
1835:
1826:
1818:
1814:
1810:
1806:
1802:
1800:
1788:
1785:
1780:
1777:
1774:
1770:
1766:
1763:
1757:
1753:
1749:
1742:
1739:
1734:
1730:
1722:
1719:
1716:
1712:
1708:
1705:
1701:
1692:
1684:
1680:
1676:
1672:
1647:
1643:
1636:
1633:
1630:
1624:
1621:
1618:
1607:
1604:
1601:
1597:
1593:
1590:
1586:
1579:
1576:
1573:
1569:
1564:
1561:
1556:
1553:
1550:
1546:
1540:
1537:
1534:
1529:
1526:
1523:
1519:
1512:
1509:
1506:
1500:
1494:
1486:
1482:
1458:
1450:
1446:
1436:
1423:
1415:
1407:
1404:
1401:
1393:
1387:
1384:
1381:
1375:
1372:
1364:
1360:
1356:
1351:
1347:
1343:
1333:
1330:
1327:
1323:
1314:
1295:
1287:
1283:
1273:
1271:
1252:
1226:
1203:
1197:
1192:
1185:
1182:
1179:
1173:
1168:
1164:
1159:
1153:
1150:
1147:
1142:
1139:
1136:
1128:
1125:
1121:
1117:
1113:
1109:
1106:
1096:
1093:
1090:
1086:
1082:
1077:
1073:
1049:
1044:
1037:
1034:
1031:
1025:
1020:
1016:
1011:
1005:
1000:
997:
994:
985:
981:
977:
973:
970:
960:
957:
954:
950:
946:
941:
937:
926:
897:
892:
889:
886:
882:
876:
873:
870:
865:
862:
859:
855:
851:
848:
842:
834:
830:
809:
799:
796:
791:
788:
785:
781:
777:
774:
768:
764:
760:
753:
750:
745:
741:
733:
730:
727:
723:
719:
716:
712:
703:
695:
691:
687:
683:
672:
669:
666:
662:
658:
653:
649:
640:
632:
630:
629:
626:appears as a
625:
624:finite string
621:
617:
613:
609:
605:
600:
586:
583:
580:
573:for any base
560:
557:
549:
545:
538:
518:
515:
507:
503:
496:
474:
470:
461:
457:
453:
449:
433:
428:
404:
400:
377:
373:
352:
349:
346:
324:
320:
311:
295:
273:
269:
246:
242:
232:
229:
227:
223:
220:
216:
212:
211:
206:
203:
195:
193:
187:
182:
174:
173:
172:
170:
166:
162:
154:
149:
141:
140:
139:
134:
130:
126:
110:
106:
105:
104:
91:
87:
86:
85:
79:
74:
64:
60:
59:
58:
56:
52:
51:concatenating
48:
44:
41:
38:
30:
26:
22:
3816:
3810:
3793:
3787:
3753:
3732:
3720:
3707:
3694:
3680:
3677:Masaaki Amou
3672:
3664:
3659:
3648:
3630:
3613:
3582:
3572:
3563:
3492:
3339:
3245:th mark for
3242:
3212:
3042:490050000000
2922:
2903:
2878:
2630:
2624:
2576:
2569:
2237:
1437:
1274:
924:
636:
607:
601:
450:
309:
233:
230:
225:
221:
214:
208:
204:
199:
191:
164:
160:
158:
132:
129:Barbier word
128:
124:
122:
108:
102:
89:
83:
62:
28:
24:
18:
3627:Staiger, L.
3169:0.123456789
3130:0.123456789
3054:0.123456789
3039:60499999499
1268:denote the
452:Kurt Mahler
202:real number
21:mathematics
3838:Categories
3776:1216.68204
3623:Priese, L.
3619:Calude, C.
3605:1086.11015
3541:References
620:characters
196:Properties
178:(sequence
145:(sequence
70:(sequence
3722:MathWorld
3635:CiteSeerX
3471:∞
3457:∩
3449:∈
3432:−
3424:−
3384:−
3373:×
3367:−
3256:⩾
3185:¯
3178:…
3141:−
3116:−
3089:∞
3074:∑
3065:−
3000:¯
2997:123456790
2966:∞
2951:∑
2893:aperiodic
2891:) and is
2885:terminate
2832:…
2821:−
2813:×
2795:−
2769:−
2761:×
2743:−
2717:−
2709:×
2691:−
2668:−
2589:δ
2546:−
2494:δ
2490:−
2461:−
2419:−
2394:−
2375:−
2339:−
2313:∞
2298:∑
2294:−
2275:−
2193:δ
2189:−
2165:−
2129:−
2110:−
2090:δ
2086:−
2062:−
2031:−
2020:−
2012:−
1958:∞
1936:∑
1932:−
1907:∞
1897:−
1879:∑
1862:−
1854:−
1815:δ
1811:−
1778:−
1767:−
1740:−
1720:−
1702:∑
1681:δ
1677:−
1634:−
1622:−
1605:−
1577:−
1562:ℓ
1554:−
1551:ℓ
1538:−
1524:ℓ
1520:∑
1510:−
1483:δ
1447:δ
1405:−
1385:−
1376:−
1339:∞
1324:∑
1284:δ
1256:⌉
1250:⌈
1230:⌋
1224:⌊
1174:
1151:−
1133:∑
1118:−
1110:⋅
1102:∞
1087:∑
1026:
991:∑
982:−
974:⋅
966:∞
951:∑
898:ℓ
890:−
887:ℓ
874:−
860:ℓ
856:∑
831:δ
789:−
778:−
751:−
731:−
713:∑
692:δ
688:−
678:∞
663:∑
628:substring
584:≥
539:μ
497:μ
350:≠
3629:(1997),
3581:(2003).
3513:See also
3275:6, 166,
2889:rational
2803:99980001
1193:⌋
1160:⌊
1045:⌉
1012:⌈
616:alphabet
612:sequence
308:for any
43:constant
3504:in the
3501:A143533
3241:in the
2915:in the
2912:A030167
2800:9999001
184:in the
181:A076478
151:in the
148:A007376
76:in the
73:A033307
55:base-10
3774:
3764:
3637:
3603:
3593:
3208:9 × 10
3019:1 × 10
2790:998001
2787:999001
2751:998001
2242:gives
1216:where
822:where
633:Series
458:. The
210:normal
23:, the
3336:, ...
3331:11111
2748:99901
35:is a
3762:ISBN
3591:ISBN
3506:OEIS
3321:1111
3279:04,
3271:are
3154:9801
2917:OEIS
2879:The
2824:2889
2738:9801
2735:9901
2699:9801
2637:are
2567:for
1242:and
1065:and
922:and
606:. A
219:base
186:OEIS
153:OEIS
123:The
103:and
78:OEIS
53:the
40:real
3821:doi
3798:doi
3772:Zbl
3601:Zbl
3334:092
3324:094
3314:096
3311:111
3304:098
3294:100
3284:102
3151:991
2772:189
2696:991
2572:≥ 2
2240:≥ 1
1474:is
1165:log
1017:log
927:− 1
618:of
489:is
462:of
171:is
127:or
19:In
3840::
3817:62
3815:,
3792:,
3770:.
3719:.
3685:,
3679:,
3625:;
3621:;
3599:.
3585:.
3549:^
3393:45
3377:10
3370:67
3364:13
3328:73
3326:,
3318:65
3316:,
3308:57
3306:,
3301:11
3298:49
3296:,
3288:41
3286:,
3281:33
3277:25
3210:.
3137:10
3103:10
3084:10
3061:10
3025:,
3023:10
2992:0.
2980:10
2944:81
2936:10
2907:10
2817:10
2765:10
2713:10
2686:81
2683:91
2663:81
2660:10
2650:10
2634:10
1315::
1272:.
883:10
835:10
782:10
765:10
742:10
724:10
696:10
684:10
654:10
599:.
519:10
508:10
475:10
448:.
405:10
378:10
247:10
200:A
136:10
80:).
66:10
32:10
3830:.
3823::
3805:.
3800::
3794:8
3778:.
3725:.
3607:.
3508:)
3479:.
3475:)
3468:,
3465:3
3461:[
3453:Z
3446:n
3443:,
3439:)
3435:2
3427:3
3421:n
3417:5
3411:n
3407:2
3402:(
3398:+
3387:3
3381:n
3358:=
3353:n
3349:d
3291:1
3259:3
3253:n
3243:n
3227:n
3223:d
3190:,
3166:=
3144:9
3133:+
3127:=
3122:)
3119:9
3113:k
3110:(
3107:2
3098:/
3094:k
3081:=
3078:k
3068:9
3057:+
3051:=
3005:,
2989:=
2984:k
2975:/
2971:k
2961:1
2958:=
2955:k
2947:=
2940:/
2919:)
2904:C
2875:.
2840:.
2836:]
2829:+
2809:)
2781:(
2777:+
2757:)
2729:(
2725:+
2720:9
2705:)
2677:(
2672:[
2655:=
2646:C
2631:C
2625:n
2610:)
2607:1
2604:+
2601:n
2598:(
2593:b
2577:n
2570:n
2553:b
2549:1
2543:b
2520:.
2515:)
2512:1
2509:+
2506:n
2503:(
2498:b
2486:b
2481:)
2473:2
2468:)
2464:1
2456:1
2453:+
2450:n
2446:b
2441:(
2435:1
2432:+
2427:n
2423:b
2414:1
2411:+
2408:n
2405:2
2401:b
2387:2
2382:)
2378:1
2370:n
2366:b
2361:(
2355:1
2352:+
2347:n
2343:b
2334:n
2331:2
2327:b
2319:(
2308:1
2305:=
2302:n
2286:2
2282:)
2278:1
2272:b
2269:(
2265:b
2260:=
2255:b
2251:C
2238:n
2219:.
2214:)
2211:1
2208:+
2205:n
2202:(
2197:b
2185:b
2177:2
2172:)
2168:1
2160:n
2156:b
2151:(
2145:1
2142:+
2137:n
2133:b
2124:n
2121:2
2117:b
2105:)
2102:n
2099:(
2094:b
2082:b
2074:2
2069:)
2065:1
2057:n
2053:b
2048:(
2042:1
2039:+
2034:1
2028:n
2024:b
2015:1
2009:n
2006:2
2002:b
1995:=
1984:)
1976:k
1973:n
1969:b
1965:k
1951:n
1947:b
1943:=
1940:k
1925:k
1922:n
1918:b
1914:k
1900:1
1894:n
1890:b
1886:=
1883:k
1874:(
1868:)
1865:1
1857:1
1851:n
1847:b
1843:(
1840:n
1836:b
1830:)
1827:n
1824:(
1819:b
1807:b
1803:=
1792:)
1789:1
1786:+
1781:1
1775:n
1771:b
1764:k
1761:(
1758:n
1754:b
1750:k
1743:1
1735:n
1731:b
1723:1
1717:n
1713:b
1709:=
1706:k
1696:)
1693:n
1690:(
1685:b
1673:b
1648:,
1644:)
1640:)
1637:b
1631:n
1628:)
1625:1
1619:b
1616:(
1613:(
1608:1
1602:n
1598:b
1594:+
1591:1
1587:(
1580:1
1574:b
1570:1
1565:=
1557:1
1547:b
1541:1
1535:n
1530:1
1527:=
1516:)
1513:1
1507:b
1504:(
1501:=
1498:)
1495:n
1492:(
1487:b
1462:)
1459:n
1456:(
1451:b
1424:.
1416:2
1412:)
1408:a
1402:1
1399:(
1394:a
1391:)
1388:1
1382:n
1379:(
1373:n
1365:n
1361:a
1357:=
1352:k
1348:a
1344:k
1334:n
1331:=
1328:k
1299:)
1296:n
1293:(
1288:b
1253:x
1227:x
1204:,
1198:)
1189:)
1186:1
1183:+
1180:k
1177:(
1169:b
1154:1
1148:n
1143:1
1140:=
1137:k
1129:+
1126:n
1122:(
1114:b
1107:n
1097:1
1094:=
1091:n
1083:=
1078:b
1074:C
1050:)
1041:)
1038:1
1035:+
1032:k
1029:(
1021:b
1006:n
1001:1
998:=
995:k
986:(
978:b
971:n
961:1
958:=
955:n
947:=
942:b
938:C
925:b
920:b
916:b
912:n
893:1
877:1
871:n
866:1
863:=
852:9
849:=
846:)
843:n
840:(
810:,
803:)
800:1
797:+
792:1
786:n
775:k
772:(
769:n
761:k
754:1
746:n
734:1
728:n
720:=
717:k
707:)
704:n
701:(
673:1
670:=
667:n
659:=
650:C
587:2
581:b
561:b
558:=
555:)
550:b
546:C
542:(
516:=
513:)
504:C
500:(
471:C
434:9
401:C
374:C
353:k
347:b
325:k
321:C
310:b
296:b
274:b
270:C
243:C
226:b
222:b
215:x
205:x
188:)
155:)
133:C
119:.
116:3
112:3
109:C
97:2
93:2
90:C
63:C
29:C
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