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 1-6 of 6 results.

A037001 Positions of the digit '2' in the decimal expansion of Pi (where positions 0, 1, 2,... refer to the digits 3, 1, 4,...).

Original entry on oeis.org

6, 16, 21, 28, 33, 53, 63, 73, 76, 83, 89, 93, 102, 112, 114, 135, 136, 140, 149, 160, 165, 173, 185, 186, 203, 221, 229, 241, 244, 260, 275, 280, 289, 292, 298, 302, 326, 329, 333, 335, 337, 354, 374, 380, 406, 423, 435, 456, 462, 477, 479, 484, 485, 500
Offset: 1

Views

Author

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

Keywords

Comments

The first few primes in this sequence are 53, 73, 83, 89, 149, 173, 229, 241, 337, 479, 571, 613, 661, 757, 829, 877, 911, 977, 991, ... - M. F. Hasler, Jul 28 2024

Links

Crossrefs

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A053746 (= a(n) + 1: the same with different offset).
Cf. A037000, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008 (similar for digits 1, ..., 9 and 0).
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.

Programs

  • Mathematica
    Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 2] (* Robert G. Wilson v, Mar 07 2011 *)
  • PARI
    A037001_upto(N=999, d=2)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 28 2024

Formula

a(n) ~ 10*n if Pi is normal, as generally assumed. - M. F. Hasler, Jul 28 2024

A053745 Positions of '1's in the decimal expansion of Pi (where positions 1,2,3,... refer to the digits 3,1,4,...).

Original entry on oeis.org

2, 4, 38, 41, 50, 69, 95, 96, 104, 111, 139, 149, 154, 155, 156, 164, 169, 175, 176, 199, 207, 221, 239, 244, 247, 251, 270, 282, 296, 298, 315, 320, 325, 343, 345, 363, 364, 382, 386, 391, 394, 396, 397, 418
Offset: 1

Views

Author

Simon Plouffe, Feb 20 2000

Keywords

Links

Crossrefs

Cf. A014976, A053746 - A053753 (the same for digits 0, ..., 9).
Cf. A088565 (primes in this sequence), A000796 (decimal digits of Pi).

Programs

  • Mathematica
    Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 1]] (* Vincenzo Librandi, Oct 07 2013 *)
  • PARI
    A053745_upto(N=444, d=1)={localprec(N+20); [i|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 29 2024, replacing earlier code from 2017

Formula

a(n) = 1 + A037000(n), a variant where position 1 is the first digit after the decimal point. - M. F. Hasler, Mar 20 2017
a(n) ~ 10*n if Pi is normal (as generally assumed but yet unproved). - M. F. Hasler, Jul 29 2024

A083609 Starting positions of strings of six 2's in the decimal expansion of Pi.

Original entry on oeis.org

963024, 1637080, 1795773, 2523356, 3474036, 5463417, 5803105, 7024615, 9742967, 11836401, 12883291, 13208202, 13371031, 15419528, 15783557, 18183625, 19081176, 20031349, 20363606, 20399387, 20735063, 21682696, 25303344, 31104717, 31614606, 32300569, 33093853, 34422277
Offset: 1

Views

Author

Rick L. Shepherd, May 01 2003

Keywords

Links

Crossrefs

Cf. A083608 (five "2"s), A118079 (seven "2"s); A037001 = A053746 - 1 (any "2"s).

Programs

  • Mathematica
    With[{s = ConstantArray[2, 6]}, SequencePosition[First@ RealDigits@ N[Pi, 10^8], s][[All, 1]] - 1] (* Michael De Vlieger, Mar 20 2017, Version 10.1 *)

Extensions

More terms from Jinyuan Wang, Feb 29 2020

A088566 Primes p such that the p-th digit in the decimal expansion of Pi is 2.

Original entry on oeis.org

7, 17, 29, 103, 113, 137, 281, 293, 457, 463, 547, 601, 631, 823, 1051, 1091, 1109, 1201, 1231, 1283, 1301, 1327, 1399, 1427, 1447, 1487, 1523, 1621, 1663, 1733, 1847, 1907, 1949, 2099, 2141, 2281, 2297, 2309, 2377, 2767, 3023, 3037, 3119, 3121, 3391, 3457
Offset: 1

Views

Author

Cino Hilliard, Nov 19 2003

Keywords

Examples

			In the decimal digits of Pi = 3.14159265... the first 2 occurs as the 7th digit, and 7 is prime; therefore a(1) = 7.

Crossrefs

Primes in A053746.
Cf. A088563 (similar for digits 0), A088565 (for digits 1),
Cf. A000796 (decimal digits of Pi).

Programs

  • PARI
    pizeros(n,d) = { default(realprecision,5000); p = Pi; v = Vec(Str(p)); for(x=1,n, if(v[x] == Str(d) && isprime(x-1),print1(x-1",")) ) }
  • PARI
    A088566_upto(N=3456, d=2)={localprec(N+20); [p|p<-primes([1, #N=digits(Pi\10^-N)]), N[p]==d]} \\ M. F. Hasler, Jul 28 2024

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 1-6 of 6 results.

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