A095765 Number of primes in range [2^n+1, 2^(n+1)] whose binary expansion begins '10' (A080165).

0, 1, 1, 3, 4, 6, 12, 22, 38, 70, 130, 237, 441, 825, 1539, 2897, 5453, 10335, 19556, 37243, 70938, 135555, 259586, 497790, 956126, 1839597, 3544827, 6839282, 13212389, 25552386, 49472951, 95883938, 186011076, 361177503, 701906519

Offset: 1

Author

Antti Karttunen, Jun 12 2004

Keywords

• nonn

Comments

I.e., number of primes p such that 2^n < p < (2^n + 2^(n-1)).

Ratio a(n)/A036378(n) converges as follows: 0, 0.5, 0.5, 0.6, 0.571429, 0.461538, 0.521739, 0.511628, 0.506667, 0.510949, 0.509804, 0.510776, 0.505734, 0.511787, 0.507921, 0.507444, 0.507303, 0.506866, 0.506173, 0.506115, 0.505487, 0.505395, 0.505318, 0.504951, 0.504786, 0.504588, 0.504437, 0.504301, 0.50415, 0.504016, 0.503887, 0.503763, 0.503654

Ratio a(n)/A095766(n) converges as follows: 0, 1, 1, 1.5, 1.333333, 0.857143, 1.090909, 1.047619, 1.027027, 1.044776, 1.04, 1.044053, 1.023202, 1.048285, 1.032193, 1.030228, 1.029645, 1.027847, 1.025001, 1.024764, 1.022191, 1.021815, 1.021501, 1.020003, 1.019331, 1.01852, 1.017908, 1.017353, 1.016737, 1.016195, 1.015669, 1.015164, 1.014723

I think this explains also the bias present in ratios shown at A095297, A095298, etc.

Examples

			Table showing the derivation of the initial terms:
  n   2^n+1  2^(n+1)  a(n)   primes starting '10' in binary
  1     3       4      0       -
  2     5       8      1       5 = 101_2
  3     9      16      1      11 = 1011_2
  4    17      32      3      17 = 10001_2, 19 = 10011_2, 23 = 10111_2
		

Links

• Antti Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
• Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Crossrefs

Cf. A036378, A080165, A095297, A095298, A095766.

Programs

Mathematica

a[n_] := PrimePi[2^n + 2^(n - 1) - 1] - PrimePi[2^n];
Array[a, 35] (* Robert G. Wilson v, Jan 24 2006 *)

Formula

a(n) = A036378(n)-A095766(n).

Extensions

a(34) and a(35) from Robert G. Wilson v, Jan 24 2006

Edited, restoring meaning of name, by Peter Munn, Jun 27 2023