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%I #27 Jul 04 2023 08:18:27
%S 0,1,1,3,4,6,12,22,38,70,130,237,441,825,1539,2897,5453,10335,19556,
%T 37243,70938,135555,259586,497790,956126,1839597,3544827,6839282,
%U 13212389,25552386,49472951,95883938,186011076,361177503,701906519
%N Number of primes in range [2^n+1, 2^(n+1)] whose binary expansion begins '10' (A080165).
%C I.e., number of primes p such that 2^n < p < (2^n + 2^(n-1)).
%C 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
%C 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
%C I think this explains also the bias present in ratios shown at A095297, A095298, etc.
%H Antti Karttunen and J. Moyer: <a href="/A095062/a095062.c.txt">C-program for computing the initial terms of this sequence</a>
%H <a href="/index/Pri#primesubsetpop2">Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]</a>
%F a(n) = A036378(n)-A095766(n).
%e Table showing the derivation of the initial terms:
%e n 2^n+1 2^(n+1) a(n) primes starting '10' in binary
%e 1 3 4 0 -
%e 2 5 8 1 5 = 101_2
%e 3 9 16 1 11 = 1011_2
%e 4 17 32 3 17 = 10001_2, 19 = 10011_2, 23 = 10111_2
%t a[n_] := PrimePi[2^n + 2^(n - 1) - 1] - PrimePi[2^n];
%t Array[a, 35] (* _Robert G. Wilson v_, Jan 24 2006 *)
%Y Cf. A036378, A080165, A095297, A095298, A095766.
%K nonn
%O 1,4
%A _Antti Karttunen_, Jun 12 2004
%E a(34) and a(35) from _Robert G. Wilson v_, Jan 24 2006
%E Edited, restoring meaning of name, by _Peter Munn_, Jun 27 2023
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