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%I #12 Jul 31 2024 08:01:34
%S 3,11,13,19,23,29,41,43,47,53,71,103,107,131,149,163,173,179,197,211,
%T 227,229,239,269,281,283,293,311,349,373,379,397,409,421,457,541,557,
%U 563,577,587,599,601,607,613,617,643,647,653,659,673,709,727,733,739,743
%N Primes indices of nonzero bits in Pi (A256108).
%C Base 2 analog of A088565 (prime positions of '1's in decimal digits of Pi).
%H Paolo Xausa, <a href="/A373471/b373471.txt">Table of n, a(n) for n = 1..10000</a>
%e Written in binary, Pi = 11.0010010000111111011010101000100010...[2] = Sum_{n >= -1} 2^-A256108(n), so the bits 1 have positions (-1, 0, 3, 6, 11, 12, 13, 14, 15, 16, 18, 19, 21, 23, 25, 29, 33, ...) and primes in this sequence are (3, 11, 13, 19, 23, ...) = this sequence.
%t Select[PositionIndex[First[RealDigits[Pi, 2, 1000]]][1] - 2, PrimeQ] (* _Paolo Xausa_, Jul 31 2024 *)
%o (PARI) select(isprime, A256108_upto(777))
%Y Cf. A256108 (indices of nonzero bits in Pi), A004601 (Pi in base 2), A051480 (run lengths in A004601).
%Y Cf. A088563 (indices of '0's in decimals of Pi).
%Y Cf. A088565 (prime indices of '1's in the decimal digits of Pi).
%K nonn,base
%O 1,1
%A _M. F. Hasler_, Jul 27 2024