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

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

1,1

Comments

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

Links

Table of n, a(n) for n=1..59.

Index entries for sequences related to the number Pi.

Formula

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

Example

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.

Mathematica

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 *)

Prog

(Python)
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)])

Crossrefs

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

Sequence in context: A199262 A382756 A158507 * A030765 A198515 A035337

Adjacent sequences: A382664 A382665 A382666 * A382668 A382669 A382670

Keyword

nonn,base

Author

James S. DeArmon, Apr 02 2025

Status

approved