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
Examples
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 ...
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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.
Programs
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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 *)
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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)])
Comments