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A036378 Number of primes p between powers of 2, 2^n < p <= 2^(n+1). 99

%I

%S 1,1,2,2,5,7,13,23,43,75,137,255,464,872,1612,3030,5709,10749,20390,

%T 38635,73586,140336,268216,513708,985818,1894120,3645744,7027290,

%U 13561907,26207278,50697537,98182656,190335585,369323305,717267168,1394192236,2712103833

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

%C 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]

%C First differences of A007053. This sequence illustrates how far the Bertrand postulate is oversatisfied.

%C Scaled for Ramanujan primes as in A190501, A190502.

%C 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]

%H Gord Palameta, <a href="/A036378/b036378.txt">Table of n, a(n) for n = 0..85</a> (using data from A007053; n = 0..74 was contributed from the same source by T. D. Noe)

%H Paul D. Beale, <a href="http://arxiv.org/abs/1411.2484">A new class of scalable parallel pseudorandom number generators based on Pohlig-Hellman exponentiation ciphers</a>, arXiv:1411.2484 [physics.comp-ph], 2014-2015.

%H Paul D. Beale, Jetanat Datephanyawat, <a href="https://arxiv.org/abs/1811.11629">Class of scalable parallel and vectorizable pseudorandom number generators based on non-cryptographic RSA exponentiation ciphers</a>, arXiv:1811.11629 [cs.CR], 2018.

%H Seung-Hoon Lee, Mario Gerla, Hugo Krawczyk, Kang-Won Lee, and Elizabeth A. Quaglia, <a href="http://www.cs.ucla.edu/~shlee/papers/netcod_TECH.pdf">Performance Evaluation of Secure Network Coding using Homomorphic Signature</a>, 2011 International Symposium on Networking Coding.

%H <a href="/index/Pri#primesubsetpop2">Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)]</a>

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

%F 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).

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

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

%o (PARI) a(n) = primepi(1<<(n+1))-primepi(1<<n)

%o (MAGMA) [1,1] cat [#PrimesInInterval(2^n, 2^(n+1)): n in [2..29]]; // _Vincenzo Librandi_, Nov 18 2014

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

%K nonn

%O 0,3

%A _Labos Elemer_

%E More terms from _Labos Elemer_, May 13 2004

%E Entries checked by _Robert G. Wilson v_, Mar 20 2006

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Last modified June 17 05:19 EDT 2019. Contains 324183 sequences. (Running on oeis4.)