cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A037000 Positions of the digit '1' in the decimal expansion of Pi.

Original entry on oeis.org

1, 3, 37, 40, 49, 68, 94, 95, 103, 110, 138, 148, 153, 154, 155, 163, 168, 174, 175, 198, 206, 220, 238, 243, 246, 250, 269, 281, 295, 297, 314, 319, 324, 342, 344, 362, 363, 381, 385, 390, 393, 395, 396, 417, 424, 427, 428, 432, 437, 438, 442, 445, 446
Offset: 1

Views

Author

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br)

Keywords

Comments

From M. F. Hasler, Jul 28 2024: (Start)
"Positions" are indices n of digits d(n) such that Pi = Sum_{n >= 0} d(n)/10^n; see A053745 for the variant where the initial digit 3 is at position 1.
The first few primes in this sequence are 3, 37, 103, 163, 269, 281, 499, 541, 547, 587, 607, 709, 797, 859, 887, 971, 983, 997, ... (End)

Links

Crossrefs

Cf. A000796 (decimals of Pi), A037001 - A037008 and A036974 (positions of other digits), A053745 (variant with all values increased by 1).

Programs

  • Maple
    P:= convert(evalf[100000](Pi),string)[3..-1]:
select(t -> P[t]="1",[$1..length(P)-1]); # Robert Israel, Dec 22 2013
  • Mathematica
    Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 1] (* Robert G. Wilson v, Mar 07 2011 *)
  • PARI
    A037000_upto(N=500, d=1)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 28 2024

Formula

Conjecturally, a(n) ~ 10n.

A096567 First digit to appear n times in the base-10 expansion of Pi.

Original entry on oeis.org

3, 1, 5, 3, 3, 3, 3, 3, 3, 9, 9, 9, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 2, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 8, 2, 2, 8, 2, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Matthew Vandermast, Jun 26 2004

Keywords

Comments

The number 7 finally appears as a(32344). - T. D. Noe, Sep 13 2012
The number 6 appears for the first time as a(99032274). - Kester Habermann, Feb 01 2021

Examples

			a(3) = 5 because 5 is the first digit to appear 3 times in the decimal expansion of Pi = 3.141(5)926(5)3(5)... - _Bobby Jacobs_, Aug 30 2017

Links

Crossrefs

Cf. A000796, A014976, A053745-A053753, A096566.
Cf. A195138, A195139, A195141, A280811.
Cf. A276686, A276992, A276993, A277171, A291599, A291600.

Programs

  • Mathematica
    nn = 1000; t = {}; d = RealDigits[Pi, 10, nn][[1]]; dCnt = Table[0, {10}]; cnt = 1; Do[b = ++dCnt[[1 + d[[n]]]]; If[b == cnt, AppendTo[t, d[[n]]]; cnt++], {n, nn}]; t (* T. D. Noe, Sep 13 2012 *)

Extensions

More terms from David Wasserman, Nov 16 2007

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

Views

Author

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

Keywords

Links

Crossrefs

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.

Programs

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

Formula

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

Extensions

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

A053746 Positions of '2's in the decimal expansion of Pi, where positions 1, 2, 3, ... correspond to digits 3, 1, 4, ...

Original entry on oeis.org

7, 17, 22, 29, 34, 54, 64, 74, 77, 84, 90, 94, 103, 113, 115, 136, 137, 141, 150, 161, 166, 174, 186, 187, 204, 222, 230, 242, 245, 261, 276, 281, 290, 293, 299, 303, 327, 330, 334, 336, 338, 355, 375, 381, 407
Offset: 1

Views

Author

Simon Plouffe, Feb 20 2000

Keywords

Comments

See A037001 for the variant where digits 3, 1, 4, ... correspond to positions 0, 1, 2, ... - M. F. Hasler, Jul 28 2024

Examples

			Pi = 3.1415926... where the first '2' occurs as the 7th digit.

Links

Crossrefs

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A037001 (= a(n) - 1: the same with different offset).
Cf. A053745 - A053753 (similar for digits 1 through 9).
Cf. A035117 (first occurrence of at least n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
Cf. A096755 (first occurrence of exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).
Cf. A121280 = A068987 - 1: position of "123...n" in Pi's decimals.
Cf. A176341: first occurrence of n in Pi's digits.
Cf. A088566 (primes in this sequence).

Programs

  • Mathematica
    Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 2]] (* Vincenzo Librandi, Oct 07 2013 *)
  • PARI
    A053746_upto(N=999)={localprec(N+20); select(d->d==2, digits(Pi\10^-N), 1)} \\ M. F. Hasler, Jul 28 2024

Formula

a(n) = A037001(n) + 1. - Georg Fischer, May 31 2021

Extensions

Changed offset from 0 to 1 by Vincenzo Librandi, Oct 07 2013

A096566 a(n) = concatenation of 10 distinct decimal digits in order of n-th appearance in the base-10 expansion of Pi.

Original entry on oeis.org

3145926870, 1539284670, 5932648170, 3295841670, 3289517460, 3958247160, 3945286017, 3942865107, 3928461057, 9382461057, 9832460517, 9832640517, 9823460517, 9823450167, 8293415067, 8923410567, 8239401567, 8249310567
Offset: 1

Views

Author

Matthew Vandermast, Jun 26 2004

Keywords

Comments

Maximal information-based nonparametric exploration statistical analysis program (MINE), which I applied "pairwise" to the first 62 terms, shows high values (up to 1) of the maximal information coefficient (MIC), which is a measure of two-variable dependence designed specifically for rapid exploration of many-dimensional data sets - see supplied links. - Alexander R. Povolotsky, Sep 07 2012
Computationally intractable question: what is the last term to appear, assuming all 9*9! terms eventually show up? - Charles R Greathouse IV, Sep 07 2012

Links

Crossrefs

Cf. A000796, A014976, A053745-A053753, A096567.

A349551 Rectangular array with ten rows, read by falling antidiagonals: row k gives positions of k in the decimal expansion (A000796) of Pi.

Original entry on oeis.org

33, 51, 2, 55, 4, 7, 66, 38, 17, 1, 72, 41, 22, 10, 3, 78, 50, 29, 16, 20, 5, 86, 69, 34, 18, 24, 9, 8, 98, 95, 54, 25, 37, 11, 21, 14, 107, 96, 64, 26, 58, 32, 23, 30, 12, 117, 104, 74, 28, 60, 49, 42, 40, 19, 6
Offset: 0

Views

Author

Clark Kimberling, Dec 17 2021

Keywords

Comments

Every positive integer occurs exactly once.
It is assumed that each digit occurs infinitely many times in A000796.

Examples

			(Base-10 digits of Pi) = (3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, ...); the position of the first 0 is 33, so the first term in row 0 is 33.
Corner:
  33, 51, 55, 66, 72, 78, 86, 98,  107,  117, 122, ... A014976
   2,  4, 38, 41, 50, 69, 95, 96,  104,  111, 139, ... A053745
   7, 17, 22, 29, 34, 54, 64, 74,   77,   84,  90, ... A053746
   1, 10, 16, 18, 25, 26, 28, 44,   47,   65,  87, ... A053747
   3, 20, 24, 37, 58, 60, 61, 71,   88,   93, 105, ... A053748
   5,  9, 11, 32, 49, 52, 62, 91,  110,  131, 132, ... A053749
   8, 21, 23, 42, 70, 73, 76, 83,   99,  109, 118, ... A053750
  14, 30, 40, 48, 57, 67, 97, 100, 121,  140, 157, ... A053751
  12, 19, 27, 35, 36, 53, 68, 75,   79,   82,  85, ... A053752
   6, 13, 15, 31, 39, 43, 45, 46,   56,   59,  63, ... A053753

Links

Crossrefs

Cf. A000796, A014976, A053745-A053753, A032445 (includes column 1).

Programs

  • Mathematica
    r = RealDigits[Pi, 10, 200][[1]]
t = Table[Flatten[Position[r, n]], {n, 0, 9}]
TableForm[t]  (* A349551 array *)
Flatten[Table[t[[n - k + 1, k]], {n, 10}, {k, n, 1, -1}]] (* A349551 sequence *)

A346122 n times the n-th digit of the decimal expansion of Pi.

Original entry on oeis.org

3, 2, 12, 4, 25, 54, 14, 48, 45, 30, 55, 96, 117, 98, 135, 48, 34, 54, 152, 80, 126, 44, 138, 96, 75, 78, 216, 84, 58, 210, 279, 160, 0, 68, 280, 288, 148, 38, 351, 280, 41, 252, 387, 132, 405, 414, 141, 336, 245, 50, 0, 260, 424, 108, 0, 504, 399, 232, 531
Offset: 1

Views

Author

Harvey P. Dale, Jul 05 2021

Keywords

Examples

			The first  digit of the decimal expansion of Pi is 3, so a(1) = 1*3 = 3.
The second digit of the decimal expansion of Pi is 1, so a(2) = 2*1 = 2.
The third  digit of the decimal expansion of Pi is 4, so a(3) = 3*4 = 12.

Links

Crossrefs

Cf. A000796, A014976 (zeros), A053745 (fixed points).
Cf. A053746, A053747, A053748, A053749, A053750, A053751, A053752, A053753.

Programs

  • Mathematica
    Module[{nn=120,pid},pid=RealDigits[Pi,10,nn][[1]];Table[n pid[[n]],{n,nn}]]
  • Python
    from sympy import S
def aupton(terms):
    digits_of_pi = "0" + str(S.Pi.n(terms+1)).replace('.', '')
    return [n*int(digits_of_pi[n]) for n in range(1, terms+1)]
print(aupton(59)) # Michael S. Branicky, Jul 08 2021
Showing 1-7 of 7 results.

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