A095765 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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