A095061 -id:A095061 - cp's OEIS frontend

A036378 Number of primes p between powers of 2, 2^n < p <= 2^(n+1).

1, 1, 2, 2, 5, 7, 13, 23, 43, 75, 137, 255, 464, 872, 1612, 3030, 5709, 10749, 20390, 38635, 73586, 140336, 268216, 513708, 985818, 1894120, 3645744, 7027290, 13561907, 26207278, 50697537, 98182656, 190335585, 369323305, 717267168, 1394192236, 2712103833

Offset: 0

Labos Elemer

nonn

Number of primes whose binary order (A029837) is n+1, i.e., those with ceiling(log_2(p)) = n+1. [corrected by Jon E. Schoenfield, May 13 2018]
First differences of A007053. This sequence illustrates how far the Bertrand postulate is oversatisfied.
Scaled for Ramanujan primes as in A190501, A190502.
This sequence appears complete such that any nonnegative number can be written as a sum of distinct terms of this sequence. The sequence has been checked for completeness up to the gap between 2^46 and 2^47. Assuming that after 2^46 the formula x/log(x) is a good approximation to primepi(x), it can be proved that 2*a(n) > a(n+1) for all n >= 46, which is a sufficient condition for completeness. [Frank M Jackson, Feb 02 2012]

			The 7 primes for which A029837(p)=6 are 37, 41, 43, 47, 53, 59, 61.

Ray Chandler, Table of n, a(n) for n = 0..91 (using data from A007053; n = 0..74 by T. D. Noe, n = 75..85 by Gord Palameta, n = 86..89 by David Baugh)
Paul D. Beale, A new class of scalable parallel pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers, arXiv:1411.2484 [physics.comp-ph], 2014-2015.
Paul D. Beale and Jetanat Datephanyawat, Class of scalable parallel and vectorizable pseudorandom number generators based on non-cryptographic RSA exponentiation ciphers, arXiv:1811.11629 [cs.CR], 2018.
Seung-Hoon Lee, Mario Gerla, Hugo Krawczyk, Kang-Won Lee, and Elizabeth A. Quaglia, Performance Evaluation of Secure Network Coding using Homomorphic Signature, 2011 International Symposium on Networking Coding.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

Cf. A000720, A190501, A190502, A190568, A007053.

[1,1] cat [#PrimesInInterval(2^n, 2^(n+1)): n in [2..29]]; // Vincenzo Librandi, Nov 18 2014

t = Table[PrimePi[2^n], {n, 0, 20}]; Rest@t - Most@t (* Robert G. Wilson v, Mar 20 2006 *)

a(n) = primepi(1<<(n+1))-primepi(1<

a(n) = primepi(2^(n+1)) - primepi(2^n).

a(n) = A095005(n)+A095006(n) = A095007(n) + A095008(n) = A095013(n) + A095014(n) = A095015(n) + A095016(n) (for n > 1) = A095021(n) + A095022(n) + A095023(n) + A095024(n) = A095019(n) + A095054(n) = A095020(n) + A095055(n) = A095060(n) + A095061(n) = A095063(n) + A095064(n) = A095094(n) + A095095(n).

More terms from Labos Elemer, May 13 2004

Entries checked by Robert G. Wilson v, Mar 20 2006

A095060 Number of fibeven primes (A095080) in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 3, 3, 9, 16, 25, 50, 83, 150, 286, 540, 975, 1865, 3515, 6588, 12620, 23835, 45486, 86811, 165822, 317770, 608517, 1170182, 2254124, 4342530, 8383468, 16197159, 31335332, 60680818, 117633364, 228260489, 443281943, 861677274

Offset: 1

Antti Karttunen, Jun 01 2004

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A036378, A095061, A095062, A095067, A095080.

a(n) = A036378(n) - A095061(n) = A095062(n) + A095067(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

A095066 Number of fib001 primes (A095086) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 3, 1, 6, 9, 15, 34, 63, 114, 206, 386, 725, 1366, 2601, 4803, 9144, 17331, 33106, 63067, 121112, 233785, 447721, 860033, 1659656, 3200843, 6188292, 11966122, 23175696, 44928209, 87187514, 169331564, 329134246

Offset: 1

Antti Karttunen, Jun 01 2004

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A095061, A095065, A095067, A095069, A095086.

a(n) = A095061(n) - A095069(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

A095081 Fibodd primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with one.

Original entry on oeis.org

17, 19, 43, 53, 59, 61, 67, 101, 103, 127, 137, 163, 179, 197, 211, 229, 239, 263, 271, 281, 307, 313, 331, 347, 349, 373, 383, 389, 433, 449, 457, 467, 491, 499, 509, 569, 577, 593, 601, 619, 643, 653, 661, 677, 739, 773, 787, 797, 821, 823

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A003622. Union of A095086 and A095089. Cf. A095061, A095080.

r = Map[Fibonacci, Range[2, 12]]; Select[Prime@ Range@ 144, Last@ Flatten@ Map[Position[r, #] &, Abs@ Differences@ NestWhileList[Function[k, k - SelectFirst[Reverse@ r, # < k &]], # + 1, # > 1 &]] == 1 &] (* Michael De Vlieger, Mar 27 2016, Version 10 *)

genit(maxx)={for(n=1,maxx,q=(n-1)+(n+sqrtint(5*n^2))\2; if(isprime(q), print1(q,",")));} \\ Bill McEachen, Mar 26 2016

from sympy import fibonacci, primerange
def a(n):
    k=0
    x=0
    while n>0:
        k=0
        while fibonacci(k)<=n: k+=1
        x+=10**(k - 3)
        n-=fibonacci(k - 1)
    return x
def ok(n):
    return str(a(n))[-1]=="1"
print([n for n in primerange(1, 1001) if ok(n)]) # Indranil Ghosh, Jun 07 2017

A095069 Number of fib101 primes (A095089) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 1, 9, 10, 20, 42, 64, 126, 251, 440, 828, 1560, 2967, 5656, 10769, 20419, 39327, 74826, 143516, 276217, 531587, 1025104, 1977596, 3821827, 7396083, 14326142, 27774012, 53875302, 104653661, 203380716

Offset: 1

Antti Karttunen, Jun 01 2004

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A095061, A095066, A095089.

a(n) = A095061(n) - A095066(n).

a(34)-a(35) from Amiram Eldar, Jun 13 2024

A095291 Number of upper Wythoff primes (A095281) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 2, 1, 2, 2, 5, 9, 13, 35, 51, 93, 175, 331, 595, 1149, 2182, 4097, 7749, 14780, 28131, 53583, 102372, 196345, 375876, 722743, 1392156, 2684022, 5180823, 10008419, 19368226, 37499404, 72698062, 141064116

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

As expected, the ratio of a(n)/A036378(n) seems to approach 1-((sqrt(5)-1)/2) (= 0.381966011250...): 0, 1, 0.5, 0.4, 0.285714, 0.384615, 0.391304, 0.302326, 0.466667, 0.372263, 0.364706, 0.377155, 0.379587, 0.369107, 0.379208, 0.382204, 0.381152, 0.380039, 0.382555, 0.382287, 0.381819, 0.381677, 0.382211, 0.381283, 0.381572, 0.381858, 0.381943, 0.382013, 0.381895, 0.382035, 0.381935, 0.381947, 0.381953

Also expected, the ratio a(n)/A095061(n) seems to approach 1: 1, 0, 0, 1, 0.5, 1.25, 1.28571, 0.72222, 1.4, 0.94444, 0.88571, 0.98315, 0.99699, 0.93407, 0.98627, 0.99453, 0.98462, 0.9973, 0.99865, 1.0011, 1.00108, 0.99979, 1.00208, 0.99622, 0.99835, 1.00039, 0.99973, 1.00046, 0.99983, 1.00031, 0.99994, 0.99994, 1.00001

A. Karttunen and J. Moyer: C-program for computing the initial terms of this sequence
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]

a(n) = A036378(n)-A095290(n). Cf. A095061, A095290.

cp's OEIS Frontend

A036378 Number of primes p between powers of 2, 2^n < p <= 2^(n+1).

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A095060 Number of fibeven primes (A095080) in range ]2^n,2^(n+1)].

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A095066 Number of fib001 primes (A095086) in range ]2^n,2^(n+1)].

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A095081 Fibodd primes, i.e., primes p whose Zeckendorf-expansion A014417(p) ends with one.

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A095069 Number of fib101 primes (A095089) in range ]2^n,2^(n+1)].

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A095291 Number of upper Wythoff primes (A095281) in range ]2^n,2^(n+1)].

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