A095006 -id:A095006 - 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

A095018 a(n) is the number of primes p which have exactly n zeros and n ones when written in binary.

Original entry on oeis.org

1, 0, 2, 4, 17, 28, 189, 531, 1990, 5747, 23902, 76658, 291478, 982793, 3677580, 13214719, 49161612, 177190667, 664806798, 2443387945

Offset: 1

Antti Karttunen, Jun 01 2004

a(n) is the number of terms in A066196 which lie between 2^(2n-1) and 2^2n inclusively.

			a(1) = 1 since only 2_10 = 10_2 satisfies the criterion;
a(2) = 0 since there is no prime between 4 and 16 which meets the criterion.
The only primes in the range ]2^5,2^6[ with equal numbers of ones and zeros in their binary expansion are 37 (in binary 100101) and 41 (in binary 101011) thus a(3)=2.
a(4) = 4 since 139, 149, 163 and 197 meet the criterion; etc.

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. A066196, A095005, A095006, A095052, A095053, A280872.

f[n_] := Block[{c = 0, p = NextPrime[2^(2n -1) -1], lmt = 2^(2n)}, While[p < lmt, If[DigitCount[p, 2, 1] == n, c++]; p = NextPrime@ p]; c]; Array[f, 17] (* K. D. Bajpai and Robert G. Wilson v, Jan 10 2017 *)

from itertools import combinations
from sympy import isprime
def A095018(n): return sum(1 for d in combinations((1<Chai Wah Wu, Jul 18 2025

Extensions

Edited by N. J. A. Sloane, Jan 16 2017

a(18)-a(20) from Amiram Eldar, Nov 21 2020

A095005 Number of odious primes (A027697) in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 1, 2, 2, 5, 8, 19, 20, 48, 75, 160, 242, 505, 835, 1761, 2799, 5890, 10250, 20921, 36872, 74316, 134816, 267749, 492286, 977207, 1823657, 3598657, 6779899, 13336543, 25358424, 49763462, 95140695, 186504600, 358630024, 702300885, 1356118149, 2654709953, 5142968571

Offset: 1

Antti Karttunen and Labos Elemer, 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. A027697, A036378, A095006.

Table[Count[Prime@ Range[PrimePi[2^n] + 1, PrimePi[2^(n + 1) - 1]], k_ /; OddQ@ First@ DigitCount[k, 2]], {n, 24}] (* Michael De Vlieger, Feb 25 2017 *)

a(n) = #select(x->((hammingweight(x)%2)==1),primes([2^n+1,2^(n+1)])); \\ Michel Marcus, Feb 26 2017

a(n) = A036378(n) - A095006(n).

a(34)-a(38) from Amiram Eldar, Jun 20 2024

A130911 a(n) is the number of primes with odd binary weight among the first n primes minus the number with an even binary weight.

Original entry on oeis.org

1, 0, -1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 6, 5, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 21, 20, 19, 20, 19, 18, 19, 18, 19, 18, 19, 18, 19, 18, 17, 16, 15, 14, 15, 14, 15, 14, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 19, 20, 19, 20, 19, 18, 19, 20, 21, 20, 19

Offset: 1

T. D. Noe, Jun 08 2007

Prime race between evil primes (A027699) and odious primes (A027697).
Shevelev conjectures that a(n) >= 0 for n > 3. Surprisingly, the conjecture also appears to be true if we count zeros instead of ones in the binary representation of prime numbers.
The conjecture is true for primes up to at least 10^13. Mauduit and Rivat prove that half of all primes are evil. - T. D. Noe, Feb 09 2009

T. D. Noe, Table of n, a(n) for n = 1..10000
CNRS Press release, The sum of digits of prime numbers is evenly distributed, May 12, 2010.
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals Math., 171 (2010), 1591-1646.
ScienceDaily, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis, May 12, 2010.
Vladimir Shevelev, A conjecture on primes and a step towards justification, arXiv:0706.0786 [math.NT], 2007.
Vladimir Shevelev, On excess of odious primes, arXiv:0707.1761 [math.NT], 2007.

Cf. A095005, A095006.
Cf. A199399, A027697, A027698, A027699, A027700, A200244, A200245, A200246, A200247.
Cf. A156549 (race between primes having an odd/even number of zeros in binary).

cnt=0; Table[p=Prime[n]; If[EvenQ[Count[IntegerDigits[p,2],1]], cnt--, cnt++ ]; cnt, {n,10000}]
Accumulate[If[OddQ[DigitCount[#,2,1]],1,-1]&/@Prime[Range[100]]] (* Harvey P. Dale, Aug 09 2013 *)

f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==1,s++)); return(s%2)};nO=0;nE=0;forprime(p=2,520,if(f(p),nO++, nE++);an=nO-nE;print1(an,", ")) \\ Washington Bomfim, Jan 14 2011

from sympy import nextprime
from itertools import islice
def agen():
    p, evod = 2, [0, 1]
    while True:
        yield evod[1] - evod[0]
        p = nextprime(p); evod[bin(p).count('1')%2] += 1
print(list(islice(agen(), 97))) # Michael S. Branicky, Dec 21 2021

a(n) = (number of odious primes <= prime(n)) - (number of evil primes <= prime(n)).

a(n) = A200247(n) - A200246(n).

Edited by N. J. A. Sloane, Nov 16 2011

A095292 Number of A095282-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 14, 23, 47, 88, 152, 295, 540, 1004, 1933, 3572, 6805, 12909, 24461, 46767, 89481, 171327, 328638, 631302, 1215243, 2342291, 4520976, 8736608, 16899331, 32727125, 63446234, 123106396

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

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)-A095293(n). Cf. A095006.

a(n)=my(s);forprime(p=2^n+1,2^(n+1), s+=valuation(p+1, 2)%2==0); s \\ Charles R Greathouse IV, Oct 09 2013

A127977 The minimum excess in the prime race of odious primes versus evil primes in the interval (2^(n-1),2^n).

Original entry on oeis.org

0, 1, 4, 7, 13, 19, 39, 53, 104, 138, 251, 334, 590, 715, 1353, 1855, 3659, 5221, 10484, 14933, 27491, 35474, 68816, 97342, 186405, 265255

Offset: 5

Jonathan Vos Post, Jun 07 2007

Shevelev conjectures (p.2) that for all natural numbers n other than 5 and 6, the number of evil primes not exceeding n <= the number of odious primes not exceeding n. Odious primes are A027697. Evil primes are A027699.

			OdiPrimePi(x) for x >= 32 is 6, 6, 6, 6, 6, 7, 7, 7, 7, 8,.. and EviPrimePi(x) for x>=32 is 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6,...
The difference OdiPrimePi(x)-EviPrimePi(x) is 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3,.. so the minimum of the difference in the interval 2^(6-1)..2^6 is 1, yielding a(6)=1.

Vladimir Shevelev, A Conjecture on Primes and a Step towards Justification, arXiv:0706.0786 [math.NT], 2007. See table 1, p. 2.
Vladimir Shevelev, On excess of the odious primes, arXiv:0707.1761 [math.NT], 2007.

Cf. A000069, A001969, A027697, A027699, A095005, A095006.

read("transforms") ; # see oeis.org/transforms.txt
isA000069 := proc(n) type(wt(n),'odd') ; end proc;
isA027697 := proc(n) isprime(n) and isA000069(n) ; end proc:
isA027699 := proc(n) isprime(n) and not isA000069(n) ; end proc:
odiPi := proc(n) option remember; if n = 0 then 0; else an1 := procname(n-1) ; if isA027697(n) then an1+1 ; else an1 ; end if; end if; end proc:
eviPi := proc(n) option remember; if n = 0 then 0; else an1 := procname(n-1) ; if isA027699(n) then an1+1 ; else an1 ; end if; end if; end proc:
oddPi := proc(n) odiPi(n)-eviPi(n) ; end proc:
A127977 := proc(n) local a,x ; a := 2^(n+1) ; for x from 2^(n-1)+1 to 2^n-1 do a := min(a,oddPi(x)) ; end do: a; end proc:
for n from 5 do print(n,A127977(n)) ; end do; # R. J. Mathar, Sep 03 2011

wt[n_] := DigitCount[n, 2, 1];
isA000069[n_] := OddQ[wt[n]];
isA027697[n_] := PrimeQ[n] && isA000069[n];
isA027699[n_] := PrimeQ[n] && !isA000069[n];
odiPi[n_] := odiPi[n] = If[n==0, 0, an1 = odiPi[n-1]; If[isA027697[n], an1+1, an1]];
eviPi[n_] := eviPi[n] = If[n==0, 0, an1 = eviPi[n-1]; If[isA027699[n], an1+1, an1]];
oddPi[n_] := odiPi[n] - eviPi[n];
A127977[n_] := Module[{a, x}, a = 2^(n+1); For[x = 2^(n-1)+1, x <= 2^n-1, x++, a = Min[a, oddPi[x]]]; a];
Table[an = A127977[n]; Print[an]; an, {n, 5, 30}] (* Jean-François Alcover, Jan 23 2018, after R. J. Mathar *)

Published numbers corrected and checked up to n=23 by R. J. Mathar, Sep 03 2011

cp's OEIS Frontend

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

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A095018 a(n) is the number of primes p which have exactly n zeros and n ones when written in binary.

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A095005 Number of odious primes (A027697) in range ]2^n,2^(n+1)].

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A130911 a(n) is the number of primes with odd binary weight among the first n primes minus the number with an even binary weight.

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A095292 Number of A095282-primes in range ]2^n,2^(n+1)].

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PARI

A127977 The minimum excess in the prime race of odious primes versus evil primes in the interval (2^(n-1),2^n).

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