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 A036378 Number of primes p between powers of 2, 2^n < p <= 2^(n+1). 99
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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] LINKS Gord Palameta, Table of n, a(n) for n = 0..85 (using data from A007053; n = 0..74 was contributed from the same source by T. D. Noe) 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, 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. 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). EXAMPLE The 7 primes for which A029837(p)=6 are 37, 41, 43, 47, 53, 59, 61. MATHEMATICA t = Table[PrimePi[2^n], {n, 0, 20}]; Rest@t - Most@t (* Robert G. Wilson v, Mar 20 2006 *) PROG (PARI) a(n) = primepi(1<<(n+1))-primepi(1<

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Last modified May 22 19:19 EDT 2019. Contains 323481 sequences. (Running on oeis4.)