A190502 -id:A190502 - 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

A190501 Number of Ramanujan primes R_k such that 2^(n-1) < R_k <= 2^n.

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 3, 6, 10, 19, 33, 62, 118, 208, 409, 740, 1418, 2676, 5043, 9638, 18248, 34949, 66752, 127880, 245489, 472113, 908302, 1751624, 3381546, 6534616, 12645372, 24490255, 47485123, 92152929, 178987716, 347943866, 676925069, 1317911597, 2567659990, 5005877954, 9765539069, 19062301793, 37230980158, 72756216207, 142253989491, 278275735952, 544621563320, 1066382258001

Offset: 0

John W. Nicholson, May 11 2011

nonn

Dana Jacobsen, Table of n, a(n) for n = 0..56

Cf. A007053, A036378, A104272, A190502.

\\ With RR[.] is a list of A104272(.). The output of this program is (for n>=1) n, A190502(n), and RR[a(n)], a(n).
j=0;while(2^j

use ntheory ":all"; say "$ ",ramanujan_prime_count(1 << $) - ramanujan_prime_count(1 << ($-1)) for 0..56; # _Dana Jacobsen, May 05 2017

Extended by T. D. Noe, May 11 2011
Modified the name as to match offset to A190502 and added leading term, John W. Nicholson, May 12 2011
Extended to n = 32 by John W. Nicholson, Dec 01 2012
Extended to n = 47, using A190502 data, by John W. Nicholson, Jan 31 2016

A181671 Number of Ramanujan primes less than 10^n.

Original entry on oeis.org

1, 10, 72, 559, 4459, 36960, 316066, 2760321, 24491666, 220098288, 1998400235, 18299775876, 168773875190, 1566017986235, 14606736768049, 136860923837558, 1287462389890262

Offset: 1

T. D. Noe, Nov 18 2010

Cf. A104272 (Ramanujan primes).

Cf. A190502 (Number of Ramanujan primes <= 2^n).

nn=50000; t=Table[0,{nn}]; s=0; Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s

use ntheory ":all"; for my $e (1..9) { say "$e ",scalar(@{ramanujan_primes(10**$e)}); } # Dana Jacobsen, May 10 2015
# To control memory use at cost of speed:

use ntheory ":all"; my($n,$inc,$start,$sum)=(1e10,1e9,0,0); while ($start < $n) { $sum += scalar(@{ramanujan_primes($start,$start+$inc-1)}); $start += $inc; } say $sum; # Dana Jacobsen, May 10 2015

use ntheory ":all"; say ramanujan_prime_count(10**$) for 1..11; # _Dana Jacobsen, Jan 03 2016

a(10)-a(11) from Dana Jacobsen, Dec 29 2014
a(12) from Dana Jacobsen, Sep 08 2015
a(13)-a(14) from Dana Jacobsen, Jan 03 2016
a(15)-a(17) from Dana Jacobsen, Apr 26 2017

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

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A190501 Number of Ramanujan primes R_k such that 2^(n-1) < R_k <= 2^n.

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A181671 Number of Ramanujan primes less than 10^n.

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