A178707 -id:A178707 - cp's OEIS frontend

A332929 Position where the binary expansion of n occurs for the first time in the binary expansion of Pi.

3, 1, 2, 1, 2, 18, 1, 13, 8, 2, 21, 18, 1, 17, 16, 13, 8, 27, 2, 62, 25, 21, 18, 93, 49, 1, 20, 17, 95, 16, 15, 13, 97, 8, 27, 45, 2, 128, 62, 146, 25, 60, 21, 395, 229, 18, 93, 209, 49, 65, 1, 78, 42, 20, 17, 105, 95, 116, 186, 16, 175, 15, 14, 13, 97, 110

Offset: 0

Thomas König, Mar 02 2020

			In binary, Pi = 11.00100100.... The bitstring 10 (for 2) occurs at position 2, so a(2) = 2.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A004601, A014777, A178707.

Cf. A032445 (for decimal expansion rather than binary).

p = RealDigits[Pi,2,500][[1]]; L = {}; Do[t = SequencePosition[p, IntegerDigits[n, 2], 1]; If[t == {}, Break[], AppendTo[L, t[[1, 1]]]], {n, 0, 65}]; L (* Giovanni Resta, Mar 16 2020 *)
Module[{nn=500,bp},bp=RealDigits[Pi,2,nn][[1]];Table[ SequencePosition[ bp,IntegerDigits[n,2],1][[All,1]],{n,0,70}]]//Flatten (* Harvey P. Dale, Sep 18 2021 *)

#! /usr/bin/perl
# Feed b004601.txt to this to get the binary digits of Pi.
while (<>) {
    chomp;
    (undef, $d[$n++]) = split(" ");
}
$pi = join("",@d);
$k = 0;
while (1) {
    last if ($pos = index($pi, sprintf("%b", $k++))) < 0;
    $out .= $pos +2 . ", ";
}
print $out,"\n";

A382667 Position of the first instance of prime(n), in base 2, in the binary representation of Pi after the binary point.

Original entry on oeis.org

3, 11, 16, 11, 16, 15, 25, 60, 91, 14, 11, 126, 58, 393, 207, 18, 14, 13, 6, 180, 141, 169, 58, 243, 47, 326, 168, 475, 15, 291, 451, 108, 64, 87, 327, 421, 358, 41, 356, 468, 343, 16, 618, 107, 80, 179, 57, 206, 291, 325, 361, 205, 427, 12, 95, 108, 436, 6, 996

Offset: 1

James S. DeArmon, Apr 02 2025

Positions are numbered starting from 1 for the first bit after the binary point in Pi.

			For n=19, the bits of Pi and their numbering, after the binary point, begin
          1 2 3 4 5 6 7 8 9 ...
   1 1 .  0 0 1 0 0 1 0 0 0 0 1 1 1 1 ...
                    \-----------/
                    prime(19) = 67
prime(19) = 1000011_2 begins at position a(19) = 6.
prime(58) = 271 = 100001111_2 also starts at 6 => a(58) = 6.

Cf. A000040, A004601, A178707, A233836, A378472, A382307.

p=Drop[RealDigits[Pi,2,1010][[1]],2](* increase for n>73 *);a[n_]:=First[SequencePosition[p,IntegerDigits[Prime[n],2]][[1]]] (* James C. McMahon, Apr 26 2025 *)

import gmpy2
from sympy import isprime
gmpy2.get_context().precision = 12000000
gmpy2.get_context().round = gmpy2.RoundDown
pi = gmpy2.const_pi()
binary_pi = gmpy2.digits(pi, 2)[0][2:] # Get binary digits and remove "11"
print([binary_pi.find(bin(cand)[2:])+1 for cand in range(2, 700) if isprime(cand)])

a(n) = A178707(A000040(n)). - Pontus von Brömssen, Apr 12 2025

A383502 Position of the first instance of prime(n), in base 3, in the ternary representation of Pi after the ternary point.

Original entry on oeis.org

2, 4, 6, 2, 12, 20, 6, 10, 27, 16, 85, 3, 35, 78, 95, 6, 38, 96, 19, 27, 9, 66, 157, 171, 81, 191, 12, 127, 52, 3, 36, 275, 88, 589, 283, 40, 290, 952, 47, 10, 1213, 750, 572, 84, 126, 2, 282, 162, 125, 480, 26, 66, 185, 157, 1490, 1310, 832, 1321, 352

Offset: 1

James S. DeArmon, Apr 28 2025

Positions are numbered starting from 1 for the first ternary digit after the ternary point in Pi.

			The ternary digits of Pi and their numbering, after the ternary point, begin
          1 2 3 4 5 6 7 8 9 ...
   0 1  . 0 2 1 1 0 1 2 2 2 2 0 1 0 2 ...
                    \---/
prime(7) is 17, or 122_3, which first appears at position 6.

Cf. A000040, A178707, A007089, A004602, A004601.

import gmpy2
from sympy import isprime
def to_base3(n):
    if n == 0: return '0'
    d = []
    while n: d.append(str(n % 3)); n //= 3
    return ''.join(reversed(d))
gmpy2.get_context().precision = 1200000
pi_ternary = gmpy2.digits(gmpy2.const_pi(),3)[0][4:]  # skip "10."
out = []
for p in range(2, 280):
    if isprime(p):
        pos = pi_ternary.find(to_base3(p)) + 1
        out.append(pos)
print(out)

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A332929 Position where the binary expansion of n occurs for the first time in the binary expansion of Pi.

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A382667 Position of the first instance of prime(n), in base 2, in the binary representation of Pi after the binary point.

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A383502 Position of the first instance of prime(n), in base 3, in the ternary representation of Pi after the ternary point.

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Python