A095055 -id:A095055 - 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

A095075 Primes in whose binary expansion the number of 1-bits is less than or equal to number of 0-bits.

Original entry on oeis.org

2, 17, 37, 41, 67, 73, 97, 131, 137, 139, 149, 163, 193, 197, 257, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 521, 523, 541, 547, 557, 563, 569, 577, 587, 593, 601, 613, 617, 641, 643, 647, 653, 659, 661, 673, 677, 709, 769, 773, 787

Offset: 1

Antti Karttunen, Jun 01 2004

			From _Indranil Ghosh_, Feb 03 2017: (Start)
17 is in the sequence because 17_10 = 10001_2. '10001' has two 1's and three 0's.
37 is in the sequence because 37_10 = 100101_2. '100101' has three 1's and 3 0's. (End)

Indranil Ghosh, Table of n, a(n) for n = 1..25000
Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.

Complement of A095070 in A000040.

Cf. A095055.

Select[Prime[Range[150]], Differences[DigitCount[#, 2]][[1]] >= 0 &] (* Amiram Eldar, Jul 25 2023 *)
Select[Prime[Range[150]],DigitCount[#,2,1]<=DigitCount[#,2,0]&] (* Harvey P. Dale, Sep 27 2023 *)

B(x) = {nB = floor(log(x)/log(2)); z1 = 0; z0 = 0;
for(i = 0, nB, if(bittest(x,i), z1++;, z0++;); );
if(z1 <= z0, return(1);, return(0););};
forprime(x = 2, 787, if(B(x), print1(x, ", "); ); );
\\ Washington Bomfim, Jan 11 2011

from sympy import isprime
i=1
j=1
while j<=250:
    if isprime(i) and bin(i)[2:].count("1")<=bin(i)[2:].count("0"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

A095328 Number of A095318-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 3, 1, 4, 13, 32, 35, 96, 124, 335, 466, 1116, 1717, 4371, 6380, 15490, 23904, 58041, 88200, 209875, 331769, 795599, 1258386, 2951789, 4741344, 11144763, 17964801, 41781268, 68371012, 158643268

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 0, 0, 0, 0.2, 0.428571, 0.076923, 0.173913, 0.302326, 0.426667, 0.255474, 0.376471, 0.267241, 0.384174, 0.289082, 0.368317, 0.300753, 0.406642, 0.312898, 0.400932, 0.324844, 0.413586, 0.328839, 0.408549, 0.336542, 0.420036, 0.345166, 0.420047, 0.349607, 0.425255, 0.354353, 0.425546, 0.359213, 0.429551

Ratios a(n)/A095055(n) converge as: 1, 1, 1, 1, 1.5, 0.333333, 0.571429, 1.181818, 1.185185, 0.875, 1.2, 0.953846, 0.976676, 0.966805, 0.945763, 0.97779, 0.977197, 0.98472, 1.006694, 0.995088, 0.988538, 0.987616, 0.983496, 0.990015, 0.991634, 0.994496, 0.995506, 0.991599, 0.996345, 0.993681, 0.993649, 0.995067, 0.995042

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)-A095329(n).

cp's OEIS Frontend

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

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A095075 Primes in whose binary expansion the number of 1-bits is less than or equal to number of 0-bits.

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Mathematica

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A095328 Number of A095318-primes in range ]2^n,2^(n+1)].

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