A095019 -id:A095019 - 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

A095071 Zero-bit dominant primes, i.e., primes whose binary expansion contains more 0's than 1's.

Original entry on oeis.org

17, 67, 73, 97, 131, 137, 193, 257, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 521, 523, 547, 577, 593, 641, 643, 673, 769, 773, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1091, 1093, 1097, 1109, 1123, 1129, 1153, 1163, 1171

Offset: 1

Antti Karttunen, Jun 01 2004

			73 is in the sequence because 73 is a prime and 73_10 = 1001001_2. '1001001' has four 0's and one 1. - _Indranil Ghosh_, Jan 31 2017

Indranil Ghosh, Table of n, a(n) for n = 1..20000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Complement of A095074 in A000040. Subset: A095072. Cf. A095019.

Reap[Do[p=Prime[k];id=IntegerDigits[p,2];n=Length@id;If[Count[id,0]>n/2,Sow[p]],{k,200}]][[2,1]]
(* Zak Seidov *)
Select[Prime[Range[200]],DigitCount[#,2,0]>DigitCount[#,2,1]&] (* Harvey P. Dale, Nov 28 2024 *)

B(x) = { nB = floor(log(x)/log(2)); b1 = 0; b0 = 0;
for(i = 0, nB, if(bittest(x,i), b1++;, b0++;); );
if(b0 > b1, return(1);, return(0););};
forprime(x = 2, 1171, if(B(x), print1(x, ", "); ); ); \\ Washington Bomfim, Jan 11 2011

{forprime(p=2,1171,nB=floor(log(p)/log(2));
sum(i=0,nB,bittest(p,i))<=nB/2&print1(p,","))} \\ Zak Seidov, Jan 11 2011

#Program to generate the b-file
from sympy import isprime
i=1
j=1
while j<=200:
    if isprime(i) and bin(i)[2:].count("0")>bin(i)[2:].count("1"):
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Jan 31 2017

A095055 Number of primes with number of 1-bits <= number of 0-bits (A095075) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 7, 11, 27, 40, 80, 130, 343, 482, 1180, 1756, 4473, 6479, 15387, 24022, 58714, 89306, 213397, 335115, 802311, 1265350, 2965114, 4781514, 11185644, 18079040, 42048314, 68709969, 159433693, 262002130, 602733406, 1000343760, 2297551889, 3828253857, 8748722270, 14683465908

Offset: 1

Antti Karttunen, Jun 01 2004

nonn

Antti Karttunen and John 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)].

Cf. A095018, A095019, A095075.

a(n) = A095019(n) + (if n is odd) A095018((n+1)/2).

a(34)-a(40) from Amiram Eldar, Jun 13 2024

A095324 Number of A095322-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 4, 13, 8, 35, 44, 124, 150, 466, 701, 1717, 2326, 6380, 9354, 23904, 34443, 88200, 134780, 331769, 508200, 1258386, 1957824, 4741344, 7424464, 17964801, 28737086, 68371012, 109643089

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 0, 0, 0, 0.2, 0, 0.076923, 0.173913, 0.302326, 0.106667, 0.255474, 0.172549, 0.267241, 0.172018, 0.289082, 0.231353, 0.300753, 0.216392, 0.312898, 0.242112, 0.324844, 0.245432, 0.328839, 0.262367, 0.336542, 0.268304, 0.345166, 0.278603, 0.349607, 0.283298, 0.354353, 0.29269, 0.359213, 0.296876

Ratios a(n)/A095019(n) converge as: 1, 1, 1, 1, 1, 0.333333, 1.333333, 1.181818, 0.8, 0.875, 0.846154, 0.953846, 0.974026, 0.966805, 1.080123, 0.97779, 0.93677, 0.98472, 0.970332, 0.995088, 0.9894, 0.987616, 0.985673, 0.990015, 0.994846, 0.994496, 0.987642, 0.991599, 0.988865, 0.993681, 0.996653, 0.995067, 0.994296

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)-A095325(n).

cp's OEIS Frontend

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

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A095071 Zero-bit dominant primes, i.e., primes whose binary expansion contains more 0's than 1's.

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A095055 Number of primes with number of 1-bits <= number of 0-bits (A095075) in range ]2^n,2^(n+1)].

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A095324 Number of A095322-primes in range ]2^n,2^(n+1)].

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