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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.

%I A382667 #55 Apr 27 2025 08:00:23
%S A382667 3,11,16,11,16,15,25,60,91,14,11,126,58,393,207,18,14,13,6,180,141,
%T A382667 169,58,243,47,326,168,475,15,291,451,108,64,87,327,421,358,41,356,
%U A382667 468,343,16,618,107,80,179,57,206,291,325,361,205,427,12,95,108,436,6,996
%N A382667 Position of the first instance of prime(n), in base 2, in the binary representation of Pi after the binary point.
%C A382667 Positions are numbered starting from 1 for the first bit after the binary point in Pi.
%F A382667 a(n) = A178707(A000040(n)). - _Pontus von Brömssen_, Apr 12 2025
%e A382667 For n=19, the bits of Pi and their numbering, after the binary point, begin
%e A382667           1 2 3 4 5 6 7 8 9 ...
%e A382667    1 1 .  0 0 1 0 0 1 0 0 0 0 1 1 1 1 ...
%e A382667                     \-----------/
%e A382667                     prime(19) = 67
%e A382667 prime(19) = 1000011_2 begins at position a(19) = 6.
%e A382667 prime(58) = 271 = 100001111_2 also starts at 6 => a(58) = 6.
%t A382667 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 *)
%o A382667 (Python)
%o A382667 import gmpy2
%o A382667 from sympy import isprime
%o A382667 gmpy2.get_context().precision = 12000000
%o A382667 gmpy2.get_context().round = gmpy2.RoundDown
%o A382667 pi = gmpy2.const_pi()
%o A382667 binary_pi = gmpy2.digits(pi, 2)[0][2:] # Get binary digits and remove "11"
%o A382667 print([binary_pi.find(bin(cand)[2:])+1 for cand in range(2, 700) if isprime(cand)])
%Y A382667 Cf. A000040, A004601, A178707, A233836, A378472, A382307.
%K A382667 nonn,base
%O A382667 1,1
%A A382667 _James S. DeArmon_, Apr 02 2025