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

Author

Labos Elemer

Keywords

• nonn

Comments

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]

Examples

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

Links

• 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)]

Crossrefs

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

Programs

Magma

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

Mathematica

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

PARI

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

Formula

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).

Extensions

More terms from Labos Elemer, May 13 2004

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