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A095298 Sum of 1-bits between the most and least significant bits summed for all primes in range ]2^n,2^(n+1)]. 7

%I #6 Mar 31 2012 14:02:23

%S 0,1,2,8,15,30,67,154,302,611,1280,2546,5207,10447,21123,42783,85726,

%T 173102,347243,698544,1401784,2813930,5644165,11328192,22712057,

%U 45538473,91288241,182965151,366691833,734702678,1471976078,2948741819

%N Sum of 1-bits between the most and least significant bits summed for all primes in range ]2^n,2^(n+1)].

%C Ratio a(n)/A036378(n) (i.e. average number of 1-bits in range ]most significant bit,least significant bit[ of primes p which 2^n < p < 2^(n+1)) grows as: 0, 0.5, 1, 1.6, 2.142857, 2.307692, 2.913043, 3.581395, 4.026667, 4.459854, 5.019608, 5.487069, 5.97133, 6.480769, 6.971287, 7.493957, 7.975254, 8.489554, 8.987783, 9.492893, 9.98877, 10.491283, 10.987107, 11.49116, 11.990823, 12.490859, 12.990533, 13.491108, 13.991985, 14.491881, 14.992221, 15.492331, 15.992713.

%C Ratio of that average compared to (n-1)/2 (the expected value of that same sum computed for all odd numbers in the same range) converges as: 1, 1, 1, 1.066667, 1.071429, 0.923077, 0.971014, 1.023256, 1.006667, 0.991079, 1.003922, 0.997649, 0.995222, 0.997041, 0.995898, 0.999194, 0.996907, 0.998771, 0.998643, 0.999252, 0.998877, 0.99917, 0.998828, 0.999231, 0.999235, 0.999269, 0.999272, 0.999341, 0.999427, 0.99944, 0.999481, 0.999505, 0.999545.

%H A. Karttunen and J. Moyer: <a href="/A095062/a095062.c.txt">C-program for computing the initial terms of this sequence</a>

%e a(1)=0, as only prime in range ]2,4] is 3, 11 in binary which has no space between its most and least significant bits. a(2)=1, as in that range there are two primes 5 (101 in binary) and 7 (111 in binary) and summing their middle bits we get 1. a(3)=2, as there are again two primes, 11 (1011 in binary) and 13 (1101 in binary) and summing the bits in the middle we get total 2.

%Y A095297, A095334. Cf. also A095353 (similar sums and ratios computed in Fibonacci number system).

%K nonn

%O 1,3

%A _Antti Karttunen_, Jun 04 2004

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