2244:
1674:
2879:
2871:
3215:
2239:{\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}}}
3039:
2541:
3224:
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
2934:
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
2256:
1225:
3210:{\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}}}
3500:
1074:
831:
1669:
2639:. 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
1079:
457:
1445:
3026:
3354:
943:
919:
655:
1488:
3504:
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
2536:{\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)}.}
3044:
1679:
242:
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.
2861:
224:
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
1251:
2631:
2576:
1277:
540:
178:) 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
1483:
1320:
582:
3280:
3032:. 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
2874:
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.
3028:
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
608:
374:
1329:
498:
428:
401:
270:
3250:
2942:
348:
297:
317:
3516:
2927:
433:
196:
163:
88:
3776:
3605:
836:
3864:
1220:{\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)},}
2651:
3495:{\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).}
2906:(because it is not an irreducible quadratic). A simple continued fraction is a continued fraction where the denominator is 1. The
1069:{\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)}}
2910:
expansion of
Champernowne's constant exhibits extremely large terms appearing between many small ones. For example, in base 10,
826:{\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)}}},}
3854:
3529:
3544:
1664:{\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),}
3869:
3799:
1280:
626:
3768:
3597:
2936:
634:
3697:
272:
is normal in base 10, while Nakai and
Shiokawa proved a more general theorem, a corollary of which is that
3859:
3645:
3351:
whose pattern becomes obvious starting with the 6th high-water mark. The number of terms can be given by
630:
470:
466:
229:
53:
47:
3747:
Sikora, J. K. "On the High Water Mark
Convergents of Champernowne's Constant in Base Ten." 3 Oct 2012.
1230:
203:
where spaces (otherwise to be ignored) have been inserted just to show the strings being concatenated.
2594:
2546:
3760:
3692:
1256:
614:
503:
1452:
1289:
3650:
2939:
of the
Champernowne constant. For example, truncating just before the 4th partial quotient, gives
545:
430:
is 0.123456789101112131415161718192021222324252627282930313. When we express this in base 9 we get
3687:
3259:
2878:
3794:
3633:
3629:
2907:
2903:
2895:
2891:
1286:
Returning to the first of these series, both the summand of the outer sum and the expression for
57:
56:
whose decimal expansion has important properties. It is named after economist and mathematician
3772:
3724:
3601:
2883:
587:
353:
3831:
3822:
3808:
3782:
3611:
3538:
1323:
476:
406:
379:
248:
17:
3228:
326:
275:
3786:
3727:
3615:
3589:
2899:
649:
3820:
Nakai, Y.; Shiokawa, I. (1992), "Discrepancy estimates for a class of normal numbers",
3711:
Analysis of the High Water Mark
Convergents of Champernowne's Constant in Various Bases
3637:
302:
179:
2870:
95:
Champernowne constants can also be constructed in other bases similarly; for example,
3848:
921:
is the number of digits between the decimal point and the first contribution from an
220:
61:
2882:
The first 161 quotients of the continued fraction of the
Champernowne constant on a
3533:
3547:, another number obtained through concatenation a representation in a given base.
462:
452:{\displaystyle {0.10888888853823026326512111305027757201400001517660835887}_{9}}
212:
50:
31:
1440:{\displaystyle \sum _{k=n}^{\infty }ka^{k}=a^{n}{\frac {n-(n-1)a}{(1-a)^{2}}}.}
3812:
3767:. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge:
3693:
Approximation to certain transcendental decimal fractions by algebraic numbers
3021:{\displaystyle 10/81=\sum _{k=1}^{\infty }k/10^{k}=0.{\overline {123456790}},}
925:-digit base-10 number; these expressions generalize to an arbitrary base
3836:
3732:
2543:
Observe that in the summand, the expression in parentheses is approximately
638:
60:, who published it as an undergraduate in 1933. The number is defined by
648:
The definition of the
Champernowne constant immediately gives rise to an
622:
914:{\displaystyle \delta _{10}(n)=9\sum _{\ell =1}^{n-1}10^{\ell -1}\ell }
65:
3225:
marks themselves grow doubly-exponentially, and the number of digits
3678:, Proc. Konin. Neder. Akad. Wet. Ser. A. 40 (1937), p. 421–428.
3509:
1, 2, 4, 18, 40, 162, 526, 1708, 4838, 13522, 34062, .... (sequence
3217:
which approximates
Champernowne's constant with error approximately
3797:(1933), "The construction of decimals normal in the scale of ten",
3710:
2877:
2869:
3748:
149:
obtained by writing it in base 10 and juxtaposing the digits:
27:
Transcendental number(s) with all positive integers in order
3759:
Cassaigne, J.; Nicolas, F. (2010). "Factor complexity". In
3676:
Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen
3511:
2922:
191:
158:
83:
3594:
Automatic Sequences: Theory, Applications, Generalizations
440:
0.10888888853823026326512111305027757201400001517660835887
3541:, another constant defined by its decimal representation
3644:, University of Auckland, New Zealand, pp. 1–35,
3357:
3262:
3231:
3042:
2945:
2654:
2597:
2549:
2259:
1677:
1491:
1455:
1332:
1292:
1259:
1233:
1082:
946:
839:
658:
590:
548:
506:
479:
436:
409:
382:
356:
329:
305:
278:
251:
3713:, in: arXiv:1408.0261, 1 Aug 2014, see Definition 9
3700:, Volume 37, Issue 2, February 1991, Pages 231–241
3494:
3274:
3244:
3209:
3020:
2855:
2625:
2570:
2535:
2238:
1663:
1477:
1439:
1314:
1271:
1245:
1219:
1068:
913:
825:
602:
576:
534:
492:
451:
422:
395:
368:
342:
311:
291:
264:
1322:can be simplified using the closed form for the
403:is normal in base 9. For example, 54 digits of
2894:expansion of Champernowne's constant does not
2856:{\displaystyle C_{10}={\frac {10}{81}}-\left.}
3663:
3532:, a similar normal number, defined using the
8:
3567:
1266:
1260:
1240:
1234:
1671:while the summand of the outer sum becomes
3800:Journal of the London Mathematical Society
3765:Combinatorics, automata, and number theory
68:representations of the positive integers:
3835:
3649:
3463:
3462:
3430:
3420:
3390:
3371:
3362:
3356:
3261:
3236:
3230:
3182:
3159:
3150:
3116:
3107:
3098:
3087:
3074:
3047:
3043:
3041:
3005:
2993:
2984:
2975:
2964:
2949:
2944:
2830:
2808:
2795:
2778:
2756:
2743:
2726:
2704:
2691:
2668:
2659:
2653:
2602:
2596:
2550:
2548:
2507:
2499:
2482:
2459:
2436:
2414:
2407:
2396:
2379:
2356:
2340:
2333:
2322:
2311:
2295:
2273:
2264:
2258:
2206:
2198:
2186:
2169:
2146:
2130:
2123:
2103:
2095:
2083:
2066:
2037:
2015:
2008:
1982:
1973:
1967:
1960:
1949:
1931:
1922:
1916:
1903:
1892:
1860:
1849:
1828:
1820:
1784:
1767:
1758:
1744:
1739:
1726:
1715:
1694:
1686:
1678:
1676:
1611:
1578:
1560:
1544:
1533:
1496:
1490:
1460:
1454:
1425:
1380:
1374:
1361:
1348:
1337:
1331:
1297:
1291:
1258:
1232:
1178:
1157:
1146:
1127:
1111:
1100:
1087:
1081:
1030:
1015:
1004:
991:
975:
964:
951:
945:
896:
880:
869:
844:
838:
795:
778:
769:
755:
750:
737:
726:
705:
697:
687:
676:
663:
657:
589:
559:
547:
517:
505:
484:
478:
443:
438:
435:
414:
408:
387:
381:
355:
334:
328:
304:
283:
277:
256:
250:
3563:
3561:
3557:
652:representation involving a double sum,
3192:90919293949596979900010203040506070809
2585:and rapidly approaches that value as
7:
3578:Cassaigne & Nicolas (2010) p.165
940:respectively. Alternative forms are
1143:
1001:
3642:Disjunctive sequences: An overview
3481:
3099:
2976:
2323:
1968:
1917:
1349:
1112:
976:
688:
376:. For example, it is not known if
25:
1246:{\displaystyle \lfloor x\rfloor }
2626:{\displaystyle \delta _{b}(n+1)}
2571:{\displaystyle {\frac {b-1}{b}}}
1324:two-dimensional geometric series
323:. It is an open problem whether
1272:{\displaystyle \lceil x\rceil }
535:{\displaystyle \mu (C_{10})=10}
239:follow a uniform distribution.
79:= 0.12345678910111213141516...
3749:http://arxiv.org/abs/1210.1263
3132:
3120:
2935:provide an exceptionally good
2620:
2608:
2525:
2513:
2292:
2279:
2224:
2212:
2115:
2109:
1878:
1853:
1840:
1834:
1802:
1771:
1706:
1700:
1650:
1638:
1626:
1623:
1526:
1514:
1508:
1502:
1478:{\displaystyle \delta _{b}(n)}
1472:
1466:
1422:
1409:
1401:
1389:
1315:{\displaystyle \delta _{b}(n)}
1309:
1303:
1199:
1187:
1051:
1039:
856:
850:
813:
782:
717:
711:
565:
552:
523:
510:
1:
2898:(because the constant is not
1449:The resulting expression for
577:{\displaystyle \mu (C_{b})=b}
142:is the sequence of digits of
3275:{\displaystyle n\geqslant 3}
3195:
3010:
2866:Continued fraction expansion
465:showed that the constant is
18:Champernowne's constant
3865:Real transcendental numbers
1281:floor and ceiling functions
929:by replacing 10 and 9 with
613:The Champernowne word is a
187:0 1 00 01 10 11 000 001 ...
154:12345678910111213141516...
3886:
3769:Cambridge University Press
3598:Cambridge University Press
2591:grows, while the exponent
3664:Nakai & Shiokawa 1992
3545:Smarandache–Wellin number
2908:simple continued fraction
2892:simple continued fraction
2633:grows exponentially with
245:Champernowne proved that
174:(sometimes also called a
106:= 0.11011100101110111...
3698:Journal of Number Theory
228:is said to be normal in
3837:10.4064/aa-62-3-271-284
3813:10.1112/jlms/s1-8.4.254
3763:; Rigo, Michel (eds.).
3728:"Champernowne constant"
3530:Copeland–Erdős constant
603:{\displaystyle b\geq 2}
369:{\displaystyle b\neq k}
3855:Mathematical constants
3496:
3276:
3246:
3211:
3103:
3022:
2980:
2887:
2875:
2857:
2627:
2572:
2537:
2327:
2240:
1972:
1921:
1757:
1665:
1555:
1479:
1441:
1353:
1316:
1273:
1247:
1221:
1168:
1116:
1070:
1020:
980:
915:
891:
827:
768:
692:
604:
578:
536:
494:
493:{\displaystyle C_{10}}
453:
424:
423:{\displaystyle C_{10}}
397:
396:{\displaystyle C_{10}}
370:
344:
313:
293:
266:
265:{\displaystyle C_{10}}
235:if its digits in base
125:= 0.12101112202122...
3588:Allouche, Jean-Paul;
3497:
3277:
3247:
3245:{\displaystyle d_{n}}
3212:
3083:
3023:
2960:
2881:
2873:
2858:
2628:
2573:
2538:
2307:
2241:
1945:
1888:
1711:
1666:
1529:
1480:
1442:
1333:
1317:
1274:
1248:
1222:
1142:
1096:
1071:
1000:
960:
916:
865:
828:
722:
672:
605:
579:
542:, and more generally
537:
495:
471:irrationality measure
454:
425:
398:
371:
345:
343:{\displaystyle C_{k}}
314:
294:
292:{\displaystyle C_{b}}
267:
172:Champernowne sequence
36:Champernowne constant
3870:Sequences and series
3771:. pp. 163–247.
3539:Liouville's constant
3355:
3260:
3229:
3186:10111213141516171819
3040:
2943:
2652:
2595:
2547:
2257:
1675:
1489:
1453:
1330:
1290:
1257:
1231:
1080:
944:
837:
656:
619:disjunctive sequence
615:disjunctive sequence
588:
546:
504:
477:
434:
407:
380:
354:
327:
303:
276:
249:
3795:Champernowne, D. G.
350:is normal in bases
3725:Weisstein, Eric W.
3492:
3272:
3242:
3207:
3205:
3018:
2888:
2876:
2853:
2623:
2568:
2533:
2236:
2234:
1661:
1475:
1437:
1312:
1269:
1243:
1217:
1066:
911:
823:
600:
574:
532:
490:
449:
420:
393:
366:
340:
309:
299:is normal in base
289:
262:
170:More generally, a
58:D. G. Champernowne
3778:978-0-521-51597-9
3607:978-0-521-82332-6
3568:Champernowne 1933
3406:
3198:
3167:
3055:
3013:
2884:logarithmic scale
2816:
2803:
2764:
2751:
2712:
2699:
2676:
2566:
2488:
2402:
2302:
2246:Summing over all
2192:
2089:
1991:
1940:
1807:
1594:
1432:
818:
633:) in which every
312:{\displaystyle b}
176:Champernowne word
136:Champernowne word
16:(Redirected from
3877:
3840:
3839:
3823:Acta Arithmetica
3815:
3790:
3751:
3745:
3739:
3738:
3737:
3720:
3714:
3709:John K. Sikora:
3707:
3701:
3685:
3679:
3674:K. Mahler,
3672:
3666:
3661:
3655:
3654:
3653:
3626:
3620:
3619:
3590:Shallit, Jeffrey
3585:
3579:
3576:
3570:
3565:
3514:
3501:
3499:
3498:
3493:
3488:
3484:
3466:
3452:
3448:
3441:
3440:
3425:
3424:
3407:
3402:
3401:
3400:
3372:
3367:
3366:
3346:
3343:
3340:
3336:
3333:
3330:
3326:
3323:
3320:
3316:
3313:
3310:
3306:
3303:
3300:
3296:
3293:
3289:
3281:
3279:
3278:
3273:
3251:
3249:
3248:
3243:
3241:
3240:
3220:
3216:
3214:
3213:
3208:
3206:
3199:
3194:
3183:
3172:
3168:
3160:
3158:
3157:
3136:
3135:
3111:
3102:
3097:
3082:
3081:
3056:
3048:
3035:
3031:
3027:
3025:
3024:
3019:
3014:
3006:
2998:
2997:
2988:
2979:
2974:
2953:
2925:
2862:
2860:
2859:
2854:
2849:
2845:
2838:
2837:
2822:
2818:
2817:
2809:
2804:
2796:
2786:
2785:
2770:
2766:
2765:
2757:
2752:
2744:
2734:
2733:
2718:
2714:
2713:
2705:
2700:
2692:
2677:
2669:
2664:
2663:
2647:
2638:
2632:
2630:
2629:
2624:
2607:
2606:
2590:
2584:
2577:
2575:
2574:
2569:
2567:
2562:
2551:
2542:
2540:
2539:
2534:
2529:
2528:
2512:
2511:
2494:
2490:
2489:
2487:
2486:
2481:
2477:
2470:
2469:
2448:
2441:
2440:
2428:
2427:
2408:
2403:
2401:
2400:
2395:
2391:
2384:
2383:
2368:
2361:
2360:
2348:
2347:
2334:
2326:
2321:
2303:
2301:
2300:
2299:
2274:
2269:
2268:
2252:
2245:
2243:
2242:
2237:
2235:
2228:
2227:
2211:
2210:
2193:
2191:
2190:
2185:
2181:
2174:
2173:
2158:
2151:
2150:
2138:
2137:
2124:
2119:
2118:
2108:
2107:
2090:
2088:
2087:
2082:
2078:
2071:
2070:
2055:
2048:
2047:
2029:
2028:
2009:
2001:
1997:
1993:
1992:
1990:
1989:
1974:
1971:
1966:
1965:
1964:
1941:
1939:
1938:
1923:
1920:
1915:
1914:
1913:
1882:
1881:
1871:
1870:
1844:
1843:
1833:
1832:
1808:
1806:
1805:
1795:
1794:
1759:
1756:
1749:
1748:
1738:
1737:
1736:
1710:
1709:
1699:
1698:
1670:
1668:
1667:
1662:
1657:
1653:
1622:
1621:
1595:
1593:
1579:
1571:
1570:
1554:
1543:
1501:
1500:
1484:
1482:
1481:
1476:
1465:
1464:
1446:
1444:
1443:
1438:
1433:
1431:
1430:
1429:
1407:
1381:
1379:
1378:
1366:
1365:
1352:
1347:
1321:
1319:
1318:
1313:
1302:
1301:
1278:
1276:
1275:
1270:
1252:
1250:
1249:
1244:
1226:
1224:
1223:
1218:
1213:
1212:
1211:
1207:
1206:
1202:
1183:
1182:
1167:
1156:
1115:
1110:
1092:
1091:
1075:
1073:
1072:
1067:
1065:
1064:
1063:
1059:
1058:
1054:
1035:
1034:
1019:
1014:
979:
974:
956:
955:
939:
932:
928:
924:
920:
918:
917:
912:
907:
906:
890:
879:
849:
848:
832:
830:
829:
824:
819:
817:
816:
806:
805:
770:
767:
760:
759:
749:
748:
747:
721:
720:
710:
709:
691:
686:
668:
667:
609:
607:
606:
601:
583:
581:
580:
575:
564:
563:
541:
539:
538:
533:
522:
521:
499:
497:
496:
491:
489:
488:
458:
456:
455:
450:
448:
447:
442:
429:
427:
426:
421:
419:
418:
402:
400:
399:
394:
392:
391:
375:
373:
372:
367:
349:
347:
346:
341:
339:
338:
318:
316:
315:
310:
298:
296:
295:
290:
288:
287:
271:
269:
268:
263:
261:
260:
194:
188:
161:
155:
129:
110:
86:
80:
45:
21:
3885:
3884:
3880:
3879:
3878:
3876:
3875:
3874:
3845:
3844:
3819:
3793:
3779:
3761:Berthé, Valérie
3758:
3755:
3754:
3746:
3742:
3723:
3722:
3721:
3717:
3708:
3704:
3686:
3682:
3673:
3669:
3662:
3658:
3628:
3627:
3623:
3608:
3600:. p. 299.
3587:
3586:
3582:
3577:
3573:
3566:
3559:
3554:
3526:
3510:
3474:
3470:
3426:
3416:
3415:
3411:
3386:
3373:
3358:
3353:
3352:
3344:
3341:
3338:
3334:
3331:
3328:
3324:
3321:
3318:
3314:
3311:
3308:
3304:
3301:
3298:
3294:
3291:
3287:
3258:
3257:
3232:
3227:
3226:
3218:
3204:
3203:
3184:
3170:
3169:
3146:
3112:
3070:
3057:
3038:
3037:
3033:
3029:
2989:
2941:
2940:
2921:
2919:
2868:
2826:
2794:
2790:
2774:
2742:
2738:
2722:
2690:
2686:
2685:
2681:
2655:
2650:
2649:
2646:
2640:
2634:
2598:
2593:
2592:
2586:
2579:
2552:
2545:
2544:
2503:
2495:
2455:
2454:
2450:
2449:
2432:
2410:
2409:
2375:
2374:
2370:
2369:
2352:
2336:
2335:
2332:
2328:
2291:
2278:
2260:
2255:
2254:
2247:
2233:
2232:
2202:
2194:
2165:
2164:
2160:
2159:
2142:
2126:
2125:
2099:
2091:
2062:
2061:
2057:
2056:
2033:
2011:
2010:
1999:
1998:
1978:
1956:
1927:
1899:
1887:
1883:
1856:
1845:
1824:
1816:
1809:
1780:
1763:
1740:
1722:
1690:
1682:
1673:
1672:
1607:
1600:
1596:
1583:
1556:
1492:
1487:
1486:
1456:
1451:
1450:
1421:
1408:
1382:
1370:
1357:
1328:
1327:
1293:
1288:
1287:
1255:
1254:
1229:
1228:
1174:
1173:
1169:
1135:
1131:
1123:
1083:
1078:
1077:
1026:
1025:
1021:
999:
995:
987:
947:
942:
941:
934:
930:
926:
922:
892:
840:
835:
834:
791:
774:
751:
733:
701:
693:
659:
654:
653:
650:infinite series
646:
625:(over a finite
621:is an infinite
586:
585:
555:
544:
543:
513:
502:
501:
480:
475:
474:
437:
432:
431:
410:
405:
404:
383:
378:
377:
352:
351:
330:
325:
324:
301:
300:
279:
274:
273:
252:
247:
246:
209:
190:
186:
157:
153:
148:
128:
124:
118:
109:
105:
99:
82:
78:
72:
44:
38:
28:
23:
22:
15:
12:
11:
5:
3883:
3881:
3873:
3872:
3867:
3862:
3857:
3847:
3846:
3843:
3842:
3830:(3): 271–284,
3817:
3807:(4): 254–260,
3791:
3777:
3753:
3752:
3740:
3715:
3702:
3680:
3667:
3656:
3651:10.1.1.34.1370
3621:
3606:
3580:
3571:
3556:
3555:
3553:
3550:
3549:
3548:
3542:
3536:
3525:
3522:
3521:
3520:
3491:
3487:
3483:
3480:
3477:
3473:
3469:
3465:
3461:
3458:
3455:
3451:
3447:
3444:
3439:
3436:
3433:
3429:
3423:
3419:
3414:
3410:
3405:
3399:
3396:
3393:
3389:
3385:
3382:
3379:
3376:
3370:
3365:
3361:
3349:
3348:
3271:
3268:
3265:
3239:
3235:
3202:
3197:
3193:
3190:
3187:
3181:
3178:
3175:
3173:
3171:
3166:
3163:
3156:
3153:
3149:
3145:
3142:
3139:
3134:
3131:
3128:
3125:
3122:
3119:
3115:
3110:
3106:
3101:
3096:
3093:
3090:
3086:
3080:
3077:
3073:
3069:
3066:
3063:
3060:
3058:
3054:
3051:
3046:
3045:
3017:
3012:
3009:
3004:
3001:
2996:
2992:
2987:
2983:
2978:
2973:
2970:
2967:
2963:
2959:
2956:
2952:
2948:
2932:
2931:
2920:= . (sequence
2917:
2867:
2864:
2852:
2848:
2844:
2841:
2836:
2833:
2829:
2825:
2821:
2815:
2812:
2807:
2802:
2799:
2793:
2789:
2784:
2781:
2777:
2773:
2769:
2763:
2760:
2755:
2750:
2747:
2741:
2737:
2732:
2729:
2725:
2721:
2717:
2711:
2708:
2703:
2698:
2695:
2689:
2684:
2680:
2675:
2672:
2667:
2662:
2658:
2644:
2622:
2619:
2616:
2613:
2610:
2605:
2601:
2565:
2561:
2558:
2555:
2532:
2527:
2524:
2521:
2518:
2515:
2510:
2506:
2502:
2498:
2493:
2485:
2480:
2476:
2473:
2468:
2465:
2462:
2458:
2453:
2447:
2444:
2439:
2435:
2431:
2426:
2423:
2420:
2417:
2413:
2406:
2399:
2394:
2390:
2387:
2382:
2378:
2373:
2367:
2364:
2359:
2355:
2351:
2346:
2343:
2339:
2331:
2325:
2320:
2317:
2314:
2310:
2306:
2298:
2294:
2290:
2287:
2284:
2281:
2277:
2272:
2267:
2263:
2231:
2226:
2223:
2220:
2217:
2214:
2209:
2205:
2201:
2197:
2189:
2184:
2180:
2177:
2172:
2168:
2163:
2157:
2154:
2149:
2145:
2141:
2136:
2133:
2129:
2122:
2117:
2114:
2111:
2106:
2102:
2098:
2094:
2086:
2081:
2077:
2074:
2069:
2065:
2060:
2054:
2051:
2046:
2043:
2040:
2036:
2032:
2027:
2024:
2021:
2018:
2014:
2007:
2004:
2002:
2000:
1996:
1988:
1985:
1981:
1977:
1970:
1963:
1959:
1955:
1952:
1948:
1944:
1937:
1934:
1930:
1926:
1919:
1912:
1909:
1906:
1902:
1898:
1895:
1891:
1886:
1880:
1877:
1874:
1869:
1866:
1863:
1859:
1855:
1852:
1848:
1842:
1839:
1836:
1831:
1827:
1823:
1819:
1815:
1812:
1810:
1804:
1801:
1798:
1793:
1790:
1787:
1783:
1779:
1776:
1773:
1770:
1766:
1762:
1755:
1752:
1747:
1743:
1735:
1732:
1729:
1725:
1721:
1718:
1714:
1708:
1705:
1702:
1697:
1693:
1689:
1685:
1681:
1680:
1660:
1656:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1620:
1617:
1614:
1610:
1606:
1603:
1599:
1592:
1589:
1586:
1582:
1577:
1574:
1569:
1566:
1563:
1559:
1553:
1550:
1547:
1542:
1539:
1536:
1532:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1499:
1495:
1474:
1471:
1468:
1463:
1459:
1436:
1428:
1424:
1420:
1417:
1414:
1411:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1377:
1373:
1369:
1364:
1360:
1356:
1351:
1346:
1343:
1340:
1336:
1311:
1308:
1305:
1300:
1296:
1268:
1265:
1262:
1242:
1239:
1236:
1216:
1210:
1205:
1201:
1198:
1195:
1192:
1189:
1186:
1181:
1177:
1172:
1166:
1163:
1160:
1155:
1152:
1149:
1145:
1141:
1138:
1134:
1130:
1126:
1122:
1119:
1114:
1109:
1106:
1103:
1099:
1095:
1090:
1086:
1062:
1057:
1053:
1050:
1047:
1044:
1041:
1038:
1033:
1029:
1024:
1018:
1013:
1010:
1007:
1003:
998:
994:
990:
986:
983:
978:
973:
970:
967:
963:
959:
954:
950:
910:
905:
902:
899:
895:
889:
886:
883:
878:
875:
872:
868:
864:
861:
858:
855:
852:
847:
843:
822:
815:
812:
809:
804:
801:
798:
794:
790:
787:
784:
781:
777:
773:
766:
763:
758:
754:
746:
743:
740:
736:
732:
729:
725:
719:
716:
713:
708:
704:
700:
696:
690:
685:
682:
679:
675:
671:
666:
662:
645:
642:
599:
596:
593:
573:
570:
567:
562:
558:
554:
551:
531:
528:
525:
520:
516:
512:
509:
487:
483:
467:transcendental
446:
441:
417:
413:
390:
386:
365:
362:
359:
337:
333:
308:
286:
282:
259:
255:
218:is said to be
208:
205:
201:
200:
180:shortlex order
168:
167:
146:
132:
131:
126:
122:
112:
111:
107:
103:
93:
92:
76:
48:transcendental
42:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3882:
3871:
3868:
3866:
3863:
3861:
3860:Number theory
3858:
3856:
3853:
3852:
3850:
3838:
3833:
3829:
3825:
3824:
3818:
3814:
3810:
3806:
3802:
3801:
3796:
3792:
3788:
3784:
3780:
3774:
3770:
3766:
3762:
3757:
3756:
3750:
3744:
3741:
3735:
3734:
3729:
3726:
3719:
3716:
3712:
3706:
3703:
3699:
3695:
3694:
3689:
3684:
3681:
3677:
3671:
3668:
3665:
3660:
3657:
3652:
3647:
3643:
3639:
3635:
3631:
3625:
3622:
3617:
3613:
3609:
3603:
3599:
3595:
3591:
3584:
3581:
3575:
3572:
3569:
3564:
3562:
3558:
3551:
3546:
3543:
3540:
3537:
3535:
3534:prime numbers
3531:
3528:
3527:
3523:
3518:
3513:
3508:
3507:
3506:
3502:
3489:
3485:
3478:
3475:
3471:
3467:
3459:
3456:
3453:
3449:
3445:
3442:
3437:
3434:
3431:
3427:
3421:
3417:
3412:
3408:
3403:
3397:
3394:
3391:
3387:
3383:
3380:
3377:
3374:
3368:
3363:
3359:
3285:
3284:
3283:
3269:
3266:
3263:
3255:
3237:
3233:
3222:
3200:
3191:
3188:
3185:
3179:
3176:
3174:
3164:
3161:
3154:
3151:
3147:
3143:
3140:
3137:
3129:
3126:
3123:
3117:
3113:
3108:
3104:
3094:
3091:
3088:
3084:
3078:
3075:
3071:
3067:
3064:
3061:
3059:
3052:
3049:
3015:
3007:
3002:
2999:
2994:
2990:
2985:
2981:
2971:
2968:
2965:
2961:
2957:
2954:
2950:
2946:
2938:
2937:approximation
2929:
2924:
2916:
2913:
2912:
2911:
2909:
2905:
2901:
2897:
2893:
2885:
2880:
2872:
2865:
2863:
2850:
2846:
2842:
2839:
2834:
2831:
2827:
2823:
2819:
2813:
2810:
2805:
2800:
2797:
2791:
2787:
2782:
2779:
2775:
2771:
2767:
2761:
2758:
2753:
2748:
2745:
2739:
2735:
2730:
2727:
2723:
2719:
2715:
2709:
2706:
2701:
2696:
2693:
2687:
2682:
2678:
2673:
2670:
2665:
2660:
2656:
2643:
2637:
2617:
2614:
2611:
2603:
2599:
2589:
2582:
2563:
2559:
2556:
2553:
2530:
2522:
2519:
2516:
2508:
2504:
2500:
2496:
2491:
2483:
2478:
2474:
2471:
2466:
2463:
2460:
2456:
2451:
2445:
2442:
2437:
2433:
2429:
2424:
2421:
2418:
2415:
2411:
2404:
2397:
2392:
2388:
2385:
2380:
2376:
2371:
2365:
2362:
2357:
2353:
2349:
2344:
2341:
2337:
2329:
2318:
2315:
2312:
2308:
2304:
2296:
2288:
2285:
2282:
2275:
2270:
2265:
2261:
2250:
2229:
2221:
2218:
2215:
2207:
2203:
2199:
2195:
2187:
2182:
2178:
2175:
2170:
2166:
2161:
2155:
2152:
2147:
2143:
2139:
2134:
2131:
2127:
2120:
2112:
2104:
2100:
2096:
2092:
2084:
2079:
2075:
2072:
2067:
2063:
2058:
2052:
2049:
2044:
2041:
2038:
2034:
2030:
2025:
2022:
2019:
2016:
2012:
2005:
2003:
1994:
1986:
1983:
1979:
1975:
1961:
1957:
1953:
1950:
1946:
1942:
1935:
1932:
1928:
1924:
1910:
1907:
1904:
1900:
1896:
1893:
1889:
1884:
1875:
1872:
1867:
1864:
1861:
1857:
1850:
1846:
1837:
1829:
1825:
1821:
1817:
1813:
1811:
1799:
1796:
1791:
1788:
1785:
1781:
1777:
1774:
1768:
1764:
1760:
1753:
1750:
1745:
1741:
1733:
1730:
1727:
1723:
1719:
1716:
1712:
1703:
1695:
1691:
1687:
1683:
1658:
1654:
1647:
1644:
1641:
1635:
1632:
1629:
1618:
1615:
1612:
1608:
1604:
1601:
1597:
1590:
1587:
1584:
1580:
1575:
1572:
1567:
1564:
1561:
1557:
1551:
1548:
1545:
1540:
1537:
1534:
1530:
1523:
1520:
1517:
1511:
1505:
1497:
1493:
1469:
1461:
1457:
1447:
1434:
1426:
1418:
1415:
1412:
1404:
1398:
1395:
1392:
1386:
1383:
1375:
1371:
1367:
1362:
1358:
1354:
1344:
1341:
1338:
1334:
1325:
1306:
1298:
1294:
1284:
1282:
1263:
1237:
1214:
1208:
1203:
1196:
1193:
1190:
1184:
1179:
1175:
1170:
1164:
1161:
1158:
1153:
1150:
1147:
1139:
1136:
1132:
1128:
1124:
1120:
1117:
1107:
1104:
1101:
1097:
1093:
1088:
1084:
1060:
1055:
1048:
1045:
1042:
1036:
1031:
1027:
1022:
1016:
1011:
1008:
1005:
996:
992:
988:
984:
981:
971:
968:
965:
961:
957:
952:
948:
937:
908:
903:
900:
897:
893:
887:
884:
881:
876:
873:
870:
866:
862:
859:
853:
845:
841:
820:
810:
807:
802:
799:
796:
792:
788:
785:
779:
775:
771:
764:
761:
756:
752:
744:
741:
738:
734:
730:
727:
723:
714:
706:
702:
698:
694:
683:
680:
677:
673:
669:
664:
660:
651:
643:
641:
640:
637:appears as a
636:
635:finite string
632:
628:
624:
620:
616:
611:
597:
594:
591:
584:for any base
571:
568:
560:
556:
549:
529:
526:
518:
514:
507:
485:
481:
472:
468:
464:
460:
444:
439:
415:
411:
388:
384:
363:
360:
357:
335:
331:
322:
306:
284:
280:
257:
253:
243:
240:
238:
234:
231:
227:
223:
222:
217:
214:
206:
204:
198:
193:
185:
184:
183:
181:
177:
173:
165:
160:
152:
151:
150:
145:
141:
137:
121:
117:
116:
115:
102:
98:
97:
96:
90:
85:
75:
71:
70:
69:
67:
63:
62:concatenating
59:
55:
52:
49:
41:
37:
33:
19:
3827:
3821:
3804:
3798:
3764:
3743:
3731:
3718:
3705:
3691:
3688:Masaaki Amou
3683:
3675:
3670:
3659:
3641:
3624:
3593:
3583:
3574:
3503:
3350:
3256:th mark for
3253:
3223:
3053:490050000000
2933:
2914:
2889:
2641:
2635:
2587:
2580:
2248:
1448:
1285:
935:
647:
618:
612:
461:
320:
244:
241:
236:
232:
225:
219:
215:
210:
202:
175:
171:
169:
143:
140:Barbier word
139:
135:
133:
119:
113:
100:
94:
73:
39:
35:
29:
3638:Staiger, L.
3180:0.123456789
3141:0.123456789
3065:0.123456789
3050:60499999499
1279:denote the
463:Kurt Mahler
213:real number
32:mathematics
3849:Categories
3787:1216.68204
3634:Priese, L.
3630:Calude, C.
3616:1086.11015
3552:References
631:characters
207:Properties
189:(sequence
156:(sequence
81:(sequence
3733:MathWorld
3646:CiteSeerX
3482:∞
3468:∩
3460:∈
3443:−
3435:−
3395:−
3384:×
3378:−
3267:⩾
3196:¯
3189:…
3152:−
3127:−
3100:∞
3085:∑
3076:−
3011:¯
3008:123456790
2977:∞
2962:∑
2904:aperiodic
2902:) and is
2896:terminate
2843:…
2832:−
2824:×
2806:−
2780:−
2772:×
2754:−
2728:−
2720:×
2702:−
2679:−
2600:δ
2557:−
2505:δ
2501:−
2472:−
2430:−
2405:−
2386:−
2350:−
2324:∞
2309:∑
2305:−
2286:−
2204:δ
2200:−
2176:−
2140:−
2121:−
2101:δ
2097:−
2073:−
2042:−
2031:−
2023:−
1969:∞
1947:∑
1943:−
1918:∞
1908:−
1890:∑
1873:−
1865:−
1826:δ
1822:−
1789:−
1778:−
1751:−
1731:−
1713:∑
1692:δ
1688:−
1645:−
1633:−
1616:−
1588:−
1573:ℓ
1565:−
1562:ℓ
1549:−
1535:ℓ
1531:∑
1521:−
1494:δ
1458:δ
1416:−
1396:−
1387:−
1350:∞
1335:∑
1295:δ
1267:⌉
1261:⌈
1241:⌋
1235:⌊
1185:
1162:−
1144:∑
1129:−
1121:⋅
1113:∞
1098:∑
1037:
1002:∑
993:−
985:⋅
977:∞
962:∑
909:ℓ
901:−
898:ℓ
885:−
871:ℓ
867:∑
842:δ
800:−
789:−
762:−
742:−
724:∑
703:δ
699:−
689:∞
674:∑
639:substring
595:≥
550:μ
508:μ
361:≠
3640:(1997),
3592:(2003).
3524:See also
3286:6, 166,
2900:rational
2814:99980001
1204:⌋
1171:⌊
1056:⌉
1023:⌈
627:alphabet
623:sequence
319:for any
54:constant
3515:in the
3512:A143533
3252:in the
2926:in the
2923:A030167
2811:9999001
195:in the
192:A076478
162:in the
159:A007376
87:in the
84:A033307
66:base-10
3785:
3775:
3648:
3614:
3604:
3219:9 × 10
3030:1 × 10
2801:998001
2798:999001
2762:998001
2253:gives
1227:where
833:where
644:Series
469:. The
221:normal
34:, the
3347:, ...
3342:11111
2759:99901
46:is a
3773:ISBN
3602:ISBN
3517:OEIS
3332:1111
3290:04,
3282:are
3165:9801
2928:OEIS
2890:The
2835:2889
2749:9801
2746:9901
2710:9801
2648:are
2578:for
1253:and
1076:and
933:and
617:. A
230:base
197:OEIS
164:OEIS
134:The
114:and
89:OEIS
64:the
51:real
3832:doi
3809:doi
3783:Zbl
3612:Zbl
3345:092
3335:094
3325:096
3322:111
3315:098
3305:100
3295:102
3162:991
2783:189
2707:991
2583:≥ 2
2251:≥ 1
1485:is
1176:log
1028:log
938:− 1
629:of
500:is
473:of
182:is
138:or
30:In
3851::
3828:62
3826:,
3803:,
3781:.
3730:.
3696:,
3690:,
3636:;
3632:;
3610:.
3596:.
3560:^
3404:45
3388:10
3381:67
3375:13
3339:73
3337:,
3329:65
3327:,
3319:57
3317:,
3312:11
3309:49
3307:,
3299:41
3297:,
3292:33
3288:25
3221:.
3148:10
3114:10
3095:10
3072:10
3036:,
3034:10
3003:0.
2991:10
2955:81
2947:10
2918:10
2828:10
2776:10
2724:10
2697:81
2694:91
2674:81
2671:10
2661:10
2645:10
1326::
1283:.
894:10
846:10
793:10
776:10
753:10
735:10
707:10
695:10
665:10
610:.
530:10
519:10
486:10
459:.
416:10
389:10
258:10
211:A
147:10
91:).
77:10
43:10
3841:.
3834::
3816:.
3811::
3805:8
3789:.
3736:.
3618:.
3519:)
3490:.
3486:)
3479:,
3476:3
3472:[
3464:Z
3457:n
3454:,
3450:)
3446:2
3438:3
3432:n
3428:5
3422:n
3418:2
3413:(
3409:+
3398:3
3392:n
3369:=
3364:n
3360:d
3302:1
3270:3
3264:n
3254:n
3238:n
3234:d
3201:,
3177:=
3155:9
3144:+
3138:=
3133:)
3130:9
3124:k
3121:(
3118:2
3109:/
3105:k
3092:=
3089:k
3079:9
3068:+
3062:=
3016:,
3000:=
2995:k
2986:/
2982:k
2972:1
2969:=
2966:k
2958:=
2951:/
2930:)
2915:C
2886:.
2851:.
2847:]
2840:+
2820:)
2792:(
2788:+
2768:)
2740:(
2736:+
2731:9
2716:)
2688:(
2683:[
2666:=
2657:C
2642:C
2636:n
2621:)
2618:1
2615:+
2612:n
2609:(
2604:b
2588:n
2581:n
2564:b
2560:1
2554:b
2531:.
2526:)
2523:1
2520:+
2517:n
2514:(
2509:b
2497:b
2492:)
2484:2
2479:)
2475:1
2467:1
2464:+
2461:n
2457:b
2452:(
2446:1
2443:+
2438:n
2434:b
2425:1
2422:+
2419:n
2416:2
2412:b
2398:2
2393:)
2389:1
2381:n
2377:b
2372:(
2366:1
2363:+
2358:n
2354:b
2345:n
2342:2
2338:b
2330:(
2319:1
2316:=
2313:n
2297:2
2293:)
2289:1
2283:b
2280:(
2276:b
2271:=
2266:b
2262:C
2249:n
2230:.
2225:)
2222:1
2219:+
2216:n
2213:(
2208:b
2196:b
2188:2
2183:)
2179:1
2171:n
2167:b
2162:(
2156:1
2153:+
2148:n
2144:b
2135:n
2132:2
2128:b
2116:)
2113:n
2110:(
2105:b
2093:b
2085:2
2080:)
2076:1
2068:n
2064:b
2059:(
2053:1
2050:+
2045:1
2039:n
2035:b
2026:1
2020:n
2017:2
2013:b
2006:=
1995:)
1987:k
1984:n
1980:b
1976:k
1962:n
1958:b
1954:=
1951:k
1936:k
1933:n
1929:b
1925:k
1911:1
1905:n
1901:b
1897:=
1894:k
1885:(
1879:)
1876:1
1868:1
1862:n
1858:b
1854:(
1851:n
1847:b
1841:)
1838:n
1835:(
1830:b
1818:b
1814:=
1803:)
1800:1
1797:+
1792:1
1786:n
1782:b
1775:k
1772:(
1769:n
1765:b
1761:k
1754:1
1746:n
1742:b
1734:1
1728:n
1724:b
1720:=
1717:k
1707:)
1704:n
1701:(
1696:b
1684:b
1659:,
1655:)
1651:)
1648:b
1642:n
1639:)
1636:1
1630:b
1627:(
1624:(
1619:1
1613:n
1609:b
1605:+
1602:1
1598:(
1591:1
1585:b
1581:1
1576:=
1568:1
1558:b
1552:1
1546:n
1541:1
1538:=
1527:)
1524:1
1518:b
1515:(
1512:=
1509:)
1506:n
1503:(
1498:b
1473:)
1470:n
1467:(
1462:b
1435:.
1427:2
1423:)
1419:a
1413:1
1410:(
1405:a
1402:)
1399:1
1393:n
1390:(
1384:n
1376:n
1372:a
1368:=
1363:k
1359:a
1355:k
1345:n
1342:=
1339:k
1310:)
1307:n
1304:(
1299:b
1264:x
1238:x
1215:,
1209:)
1200:)
1197:1
1194:+
1191:k
1188:(
1180:b
1165:1
1159:n
1154:1
1151:=
1148:k
1140:+
1137:n
1133:(
1125:b
1118:n
1108:1
1105:=
1102:n
1094:=
1089:b
1085:C
1061:)
1052:)
1049:1
1046:+
1043:k
1040:(
1032:b
1017:n
1012:1
1009:=
1006:k
997:(
989:b
982:n
972:1
969:=
966:n
958:=
953:b
949:C
936:b
931:b
927:b
923:n
904:1
888:1
882:n
877:1
874:=
863:9
860:=
857:)
854:n
851:(
821:,
814:)
811:1
808:+
803:1
797:n
786:k
783:(
780:n
772:k
765:1
757:n
745:1
739:n
731:=
728:k
718:)
715:n
712:(
684:1
681:=
678:n
670:=
661:C
598:2
592:b
572:b
569:=
566:)
561:b
557:C
553:(
527:=
524:)
515:C
511:(
482:C
445:9
412:C
385:C
364:k
358:b
336:k
332:C
321:b
307:b
285:b
281:C
254:C
237:b
233:b
226:x
216:x
199:)
166:)
144:C
130:.
127:3
123:3
120:C
108:2
104:2
101:C
74:C
40:C
20:)
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