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

1,4

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.

Links

Table of n, a(n) for n=1..35.

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)]

Formula

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

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A329671 A124571 A332279 * A095016 A358358 A160684

Adjacent sequences: A095762 A095763 A095764 * A095766 A095767 A095768

Keyword

nonn

Author

Antti Karttunen, Jun 12 2004

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

Status

approved