A037008 -id:A037008 - cp's OEIS frontend

A057680 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where the initial digit 3 is at position 0.

1, 16470, 44899, 79873884, 711939213, 36541622473, 45677255610, 62644957128, 656430109694

Offset: 1

Mike Keith, Oct 19 2000

The average number of matches of length "n" digits is exactly 0.9.  That is, we expect 0.9 matches with 1 digit, 0.9 matches with 2 digits, etc.  Increasing the number of digits by a factor of 10 means that we expect to find 0.9 new matches.  Increasing the search from 10^11 to 10^12 (which includes 10 times as much work) would thus only expect to find 0.9 new matches. - Alan Eliasen, May 01 2013 (corrected by Michael Beight, Mar 21 2020)
Consequently, with the second Borel-Cantelli lemma, the expected number of terms in this sequence is infinite with probability 1. (Of course the sequence is not random, but almost all of the sequences corresponding to randomly-chosen real numbers in place of Pi have infinitely many terms.) - Charles R Greathouse IV, Apr 29 2013
a(1) & a(5) are the first occurrences in Pi of their respective strings; a(2) & a(4) are the second occurrences; a(3) is the fourth occurrence. - Hans Havermann, Jul 27 2014
A near-miss '043611' occurs at position 43611. - S. Alwin Mao, Feb 18 2020
a(10) > 5 * 10^13. - Kang Seonghoon, Nov 02 2020
Has no terms in common with A037008 (but see Mao comment above). - Charles R Greathouse IV, Jun 21 2022

			1 is a term because the string of digits '1' occurs as the 1st digit after the decimal point.
Similarly, 16470 is a term because the string of digits '16470' occurs starting at position 16470 (after the decimal point) in the digits of Pi (although it already occurs earlier at position 1602). - _M. F. Hasler_, Jul 29 2024

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

David G. Andersen, The Pi-Search Page.
Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).
Google, 50 trillion digits of pi (2020).

Cf. A000796 (decimal digits of Pi), A057679 (variant where position 1 refers to the initial digit 3), A064810 (variant where position 0 refers to the first digit after the decimal point), A109513 (variant using chunks of m digits).

StringsinPiAfterPoint[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[Pi,10,cc] // First // Rest, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i,Length[sol]}];] (* For example, StringsinPiAfterPoint[5] returns all 5-digit members of the sequence. - Colin Rose, Mar 15 2006 *)
Do[If[RealDigits[Pi,10,a=i+IntegerLength@i-1,-1][[1,i;;a]]==IntegerDigits@i,Print@i],{i,50000}] (* Giorgos Kalogeropoulos, Feb 21 2020 *)

A057680_row(r=5)={my(M=10^r, R=[]); for(n=M\10, M-1, localprec(n+r); Pi\10^(1-r-n)%M==n && !print1(n",") && R=concat(R,n));R} \\ prints & returns the r-digit terms. - M. F. Hasler, Jul 29 2024

More terms from Colin Rose, Mar 15 2006
a(5) from Nathaniel Johnston, Nov 12 2010
a(6)-a(8) from Alan Eliasen, May 01 2013
a(9) from Alan Eliasen, Jun 06 2013
Name clarified by Kang Seonghoon, Nov 02 2020
Edited by M. F. Hasler, Jul 29 2024

A014976 Successive locations of zeros in decimal expansion of Pi (where locations 1, 2, 3, ... correspond to digits 3, 1, 4, ...).

Original entry on oeis.org

33, 51, 55, 66, 72, 78, 86, 98, 107, 117, 122, 129, 133, 147, 160, 165, 168, 177, 196, 208, 246, 249, 265, 271, 288, 292, 308, 309, 312, 328, 331, 341, 358, 361, 362, 367, 370, 376, 399, 404, 409, 422, 444, 452, 494, 514, 521, 524, 544, 546, 553, 558, 562, 597, 602, 603, 604, 619, 639, 658

Offset: 1

Bagirath R. Krishnamachari (bagi(AT)callisto.miel.mot.com)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to the number Pi

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A053745 - A053753 (similar for digits 1 through 9).
Cf. A037008 for a variant with all values decreased by 1.
See A088563 for primes in this sequence.

f := proc(n) if pi[n]=0 then n fi; end;[seq(f(i),i=1..2000)]; # where pi is an array with the digits of Pi. - Simon Plouffe [Corrected by Neven Juric, Jul 08 2008]

Flatten[ Position[ RealDigits[Pi, 10, 660] [[1]], 0]] (* Robert G. Wilson v, Mar 19 2004 *)

A014976_upto(N=999)={localprec(N+20); select(d->!d, digits(Pi\10^-N), 1)} \\ Returns a "Vecsmall": use Vec(...) if needed, or alternatively: {...; [i|i<-[1..#N=digits(Pi\10^-N)], !N[i]]}. - M. F. Hasler, Jul 28 2024

a(n) = A037008(n) + 1. - Georg Fischer, May 31 2021

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A050208 Starting positions of strings of 2 1's in the decimal expansion of Pi.

Original entry on oeis.org

94, 153, 154, 174, 362, 395, 427, 437, 445, 494, 709, 758, 846, 921, 983, 984, 1129, 1185, 1349, 1503, 1534, 1656, 1849, 1896, 2458, 2497, 2724, 2740, 2779, 2904, 3253, 3398, 3493, 3503, 3504, 3534, 3735, 3822, 3844, 3847, 3963, 3992, 3993

Offset: 1

Eric W. Weisstein

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pi Digits
Index entries for sequences related to the number Pi

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A134212, A134213, A134214, A134215, A134216, A134217, A134218, A134218, A134219.

a=RealDigits[Pi,10,8! ];lst={};Do[b=a[[1]][[n]];c=a[[1]][[n+1]];If[b==1&&c==1,AppendTo[lst,n-1]],{n,8!-1}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 02 2009 *)
Flatten[Position[Partition[RealDigits[Pi,10,4000][[1]],2,1],?(#=={1,1}&) ]]-1 (* _Harvey P. Dale, Dec 01 2013 *)
SequencePosition[RealDigits[Pi,10,4000][[1]],{1,1}][[;;,1]]-1 (* Harvey P. Dale, Jul 08 2024 *)

Edited by N. J. A. Sloane, Aug 10 2010

A134212 Positions of 12 after decimal point in decimal expansion of Pi.

Original entry on oeis.org

148, 220, 243, 297, 483, 499, 607, 660, 710, 712, 963, 1080, 1199, 1299, 1350, 1425, 1843, 1864, 1924, 2013, 2287, 2295, 2307, 2341, 2498, 2632, 2938, 2976, 3258, 3322, 3328, 3346, 3426, 3455, 3499, 3690, 3713, 3823, 3892, 4056, 4160, 4220, 4252, 4372, 4414, 4450

Offset: 1

Artur Jasinski, Oct 14 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A050208, A134213, A134214, A134215, A134216, A134217, A134218, A134218, A134219.

First /@ SequencePosition[RealDigits[Pi - 3, 10, 5000][[1]], {1, 2}] (* Amiram Eldar, Mar 20 2020 *)

More terms from Amiram Eldar, Mar 20 2020

A134213 Positions of 13 after decimal point in decimal expansion of Pi.

Original entry on oeis.org

110, 281, 363, 381, 525, 598, 627, 734, 744, 759, 859, 971, 1076, 1096, 1186, 1304, 1653, 1666, 1928, 1967, 1985, 2080, 2197, 2311, 2428, 2545, 2619, 2677, 2727, 2741, 2752, 2849, 3043, 3157, 3466, 3505, 3616, 3761, 3794, 3828, 3902, 4112, 4213, 4245, 4381

Offset: 1

Artur Jasinski, Oct 14 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000796, A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A050208, A134212, A134214, A134215, A134216, A134217, A134218, A134218, A134219.

First /@ SequencePosition[RealDigits[Pi - 3, 10, 5000][[1]], {1, 3}] (* Amiram Eldar, Mar 20 2020 *)

A134214 Positions of 14 after decimal point in decimal expansion of Pi.

Original entry on oeis.org

1, 103, 250, 295, 385, 610, 651, 670, 882, 895, 1382, 1612, 1628, 1636, 1638, 1696, 1812, 2067, 2118, 2121, 2331, 2539, 2606, 2666, 2685, 2725, 2763, 2782, 2844, 2888, 2916, 2966, 2971, 3013, 3034, 3096, 3125, 3135, 3186, 3237, 3250, 3325, 3362, 3475, 3497

Offset: 1

Artur Jasinski, Oct 14 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000796, A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A050208, A134212, A134213, A134215, A134216, A134217, A134218, A134218, A134219.

First /@ SequencePosition[RealDigits[Pi - 3, 10, 5000][[1]], {1, 4}] (* Amiram Eldar, Mar 20 2020 *)

A134215 Positions of 15 after decimal point in decimal expansion of Pi.

Original entry on oeis.org

3, 314, 324, 344, 393, 730, 922, 1030, 1098, 1100, 1114, 1315, 1342, 1436, 1657, 2148, 2150, 2215, 2327, 2389, 2501, 2565, 2688, 2957, 3000, 3093, 3099, 3275, 3280, 3354, 3414, 3464, 3522, 3532, 3553, 3644, 3858, 3959, 4362, 4389, 4536, 4597, 4645, 4828, 4866

Offset: 1

Artur Jasinski, Oct 14 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A050208, A134212, A134213, A134214, A134216, A134217, A134218, A134218, A134219.

First /@ SequencePosition[RealDigits[Pi - 3, 10, 5000][[1]], {1, 5}] (* Amiram Eldar, Mar 20 2020 *)

More terms from Amiram Eldar, Mar 20 2020

A134216 Positions of 16 after decimal point in decimal expansion of Pi.

Original entry on oeis.org

40, 68, 238, 396, 791, 992, 1130, 1182, 1206, 1324, 1410, 1422, 1504, 1602, 1610, 1651, 1709, 1767, 2142, 2185, 2345, 2352, 2402, 2459, 2640, 2746, 3024, 3220, 3370, 3399, 3409, 3516, 3529, 3535, 3585, 3650, 3660, 3860, 3978, 3994, 4128, 4342, 4385, 4648, 4653

Offset: 1

Artur Jasinski, Oct 14 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A050208, A134212, A134213, A134214, A134215, A134217, A134218, A134218, A134219.

First /@ SequencePosition[RealDigits[Pi - 3, 10, 5000][[1]], {1, 6}] (* Amiram Eldar, Mar 20 2020 *)

More terms from Amiram Eldar, Mar 20 2020

A134217 Positions of 17 after decimal point in decimal expansion of Pi.

Original entry on oeis.org

95, 138, 155, 319, 342, 428, 438, 547, 566, 568, 574, 640, 646, 786, 850, 887, 889, 961, 1086, 1134, 1152, 1419, 1577, 1621, 1625, 1758, 1788, 1850, 1946, 2109, 2187, 2491, 2573, 2591, 2947, 3177, 3310, 3393, 3419, 3539, 3789, 4094, 4218, 4418, 4430, 4444, 4510

Offset: 1

Artur Jasinski, Oct 14 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A050208, A134212, A134213, A134214, A134215, A134216, A134218, A134218, A134219.

Flatten[Position[Partition[Rest[RealDigits[\[Pi],10,3000][[1]]],2,1], {1,7}]]  (* Harvey P. Dale, Apr 07 2011 *)

More terms from Harvey P. Dale, Apr 07 2011

More terms from Amiram Eldar, Mar 20 2020

A134218 Positions of 18 after decimal point in decimal expansion of Pi.

Original entry on oeis.org

424, 446, 471, 490, 587, 728, 752, 797, 847, 951, 1035, 1056, 1225, 1444, 1539, 1572, 1574, 1674, 1715, 1738, 1897, 2220, 2628, 2776, 2867, 2942, 2964, 2989, 3082, 3149, 3318, 3339, 3343, 3439, 3549, 3596, 3607, 3664, 3801, 3922, 3956, 4044, 4070, 4145, 4179, 4485

Offset: 1

Artur Jasinski, Oct 14 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A050208, A134212, A134213, A134214, A134215, A134216, A134217, A134219.

First /@ SequencePosition[RealDigits[Pi - 3, 10, 5000][[1]], {1, 8}] (* Amiram Eldar, Mar 20 2020 *)

More terms from Amiram Eldar, Mar 20 2020

cp's OEIS Frontend

A057680 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where the initial digit 3 is at position 0.

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A014976 Successive locations of zeros in decimal expansion of Pi (where locations 1, 2, 3, ... correspond to digits 3, 1, 4, ...).

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A050208 Starting positions of strings of 2 1's in the decimal expansion of Pi.

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A134212 Positions of 12 after decimal point in decimal expansion of Pi.

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A134213 Positions of 13 after decimal point in decimal expansion of Pi.

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A134214 Positions of 14 after decimal point in decimal expansion of Pi.

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A134215 Positions of 15 after decimal point in decimal expansion of Pi.

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A134216 Positions of 16 after decimal point in decimal expansion of Pi.

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A134217 Positions of 17 after decimal point in decimal expansion of Pi.

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