cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

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Author

Antti Karttunen, Jun 04 2004

Keywords

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

Crossrefs

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

A095354 Number of primes p such that Fib(n+1) <= p < Fib(n+2), (where Fib = A000045).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 3, 5, 7, 11, 16, 24, 37, 55, 84, 126, 198, 297, 458, 704, 1087, 1674, 2602, 4029, 6263, 9738, 15186, 23705, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419527, 13298631
Offset: 1

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Author

Antti Karttunen, Jun 04 2004

Keywords

Examples

			I.e. gives the number of primes whose Zeckendorf-expansion is n fibits long. a(1) = a(2) = 0, as there are no primes in ranges [1,2[ and [2,3[. a(3)=1 as in [3,5[ there is prime 3 with Fibonacci-representation 100. a(4)=2, as in [5,8[ there are primes 5 and 7. a(5)=1, as in [8,13[ there is only one prime 11 and a(6)=3 as in [13,21[ there are primes 13,17,19.

Links

Crossrefs

Cf. A095353, A036378.

Extensions

a(2) corrected by Chai Wah Wu, Jan 16 2020

A095336 Sum of 1-fibits in Zeckendorf-expansion A014417(p) summed for all primes p in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 3, 3, 13, 20, 41, 76, 176, 325, 638, 1353, 2533, 5223, 10186, 20504, 40775, 80661, 163765, 318602, 649948, 1268922, 2571531, 5082895, 10217300, 20327307, 40399966, 82164918, 160343669, 324931245, 640501167, 1290990369, 2567150515
Offset: 1

Views

Author

Antti Karttunen, Jun 04 2004

Keywords

Examples

			a(1)=1, as only prime in range ]2,4] is 3, whose Fibonacci-representation is 100. In the next range we have primes 5 and 7, whose Fibonacci-representations are 1000 and 1010 respectively, thus a(2)=3.

Links

Crossrefs

Cf. A095298, A095353.
Showing 1-3 of 3 results.

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