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

0, 1, 2, 8, 15, 30, 67, 154, 302, 611, 1280, 2546, 5207, 10447, 21123, 42783, 85726, 173102, 347243, 698544, 1401784, 2813930, 5644165, 11328192, 22712057, 45538473, 91288241, 182965151, 366691833, 734702678, 1471976078, 2948741819

Offset: 1

Author

Antti Karttunen, Jun 04 2004

Keywords

• nonn

Comments

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.

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.

Examples

			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.
		

Links

• A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence

Crossrefs

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