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

1,3

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.

LINKS

Table of n, a(n) for n=1..32.

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

EXAMPLE

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.

CROSSREFS

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

Sequence in context: A292202 A189952 A077598 * A297734 A100596 A295937

Adjacent sequences:  A095295 A095296 A095297 * A095299 A095300 A095301

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 04 2004

STATUS

approved

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)