A095052 -id:A095052 - cp's OEIS frontend

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

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

A095072 Primes in whose binary expansion the number of 0-bits is one more than the number of 1-bits.

Original entry on oeis.org

17, 67, 73, 97, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 1039, 1051, 1063, 1069, 1109, 1123, 1129, 1163, 1171, 1187, 1193, 1201, 1249, 1291, 1301, 1321, 1361, 1543, 1549, 1571, 1609, 1667, 1669, 1697, 1801, 4127, 4157, 4211, 4217

Offset: 1

Antti Karttunen, Jun 01 2004

A010051(a(n)) = 1 and A037861(a(n)) = 1. - Reinhard Zumkeller, Mar 31 2015

			97 is in the sequence because 97 is a prime and 97_10 = 1100001_2. The number of 0's in 1100001 is 4 and the number of 1's is 3. - _Indranil Ghosh_, Jan 31 2017

Indranil Ghosh, Table of n, a(n) for n = 1..20000 (terms 1..1000 from Reinhard Zumkeller)
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A031444. Subset of A095071.
Cf. A095052.
Cf. A010051, A037861.

a095072 n = a095072_list !! (n-1)
a095072_list = filter ((== 1) . a010051' . fromIntegral) a031444_list
-- Reinhard Zumkeller, Mar 31 2015

Select[Prime[Range[500]], Differences[DigitCount[#, 2]] == {1} &]

isA095072(n)=my(v=binary(n));#v==2*sum(i=1,#v,v[i])+1&&isprime(n)

forprime(p=2, 4250, v=binary(p); s=0; for(k=1, #v, s+=if(v[k]==0,+1,-1)); if(s==1,print1(p,", ")))

#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")==1:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Jan 31 2017

A095294 Number of A095284-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 4, 8, 15, 44, 47, 150, 236, 701, 863, 2326, 3298, 9354, 12933, 34443, 51300, 134780, 199410, 508200, 769127, 1957824, 2978179, 7424464, 11590386, 28737086, 44867556, 109643089

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 0, 0, 0, 0, 0, 0.076923, 0.173913, 0.093023, 0.106667, 0.109489, 0.172549, 0.101293, 0.172018, 0.146402, 0.231353, 0.151165, 0.216392, 0.161746, 0.242112, 0.175754, 0.245432, 0.191264, 0.262367, 0.202279, 0.268304, 0.210966, 0.278603, 0.219599, 0.283298, 0.228618, 0.29269, 0.235729, 0.296876

Ratios a(n)/A095327(n) converge as: 1, 1, 1, 1, 1, 0,1.333333, 4., 0.8, 1, 0.846154, 0.903846, 0.974026, 1.18593, 1.080123,1.015294, 0.93677, 0.960116, 0.970332, 0.987101, 0.9894, 0.998326, 0.985673, 0.997384, 0.994846, 0.988856, 0.987642, 0.987035, 0.988865, 0.993762, 0.996653, 0.994302, 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)-A095295(n). Cf. also A095329, A095052, A095053.

A095295 Number of A095285-primes in range ]2^n,2^(n+1)].

Original entry on oeis.org

1, 2, 2, 5, 7, 12, 19, 39, 67, 122, 211, 417, 722, 1376, 2329, 4846, 8423, 17092, 29281, 60653, 105893, 216916, 378928, 786408, 1385920, 2876617, 5069466, 10583728, 18782814, 39107151, 69445570, 145468029, 259680216

Offset: 1

Antti Karttunen, Jun 04 2004

nonn

Ratios a(n)/A036378(n) converge as: 1, 1, 1, 1, 1, 0.923077, 0.826087, 0.906977, 0.893333, 0.890511, 0.827451, 0.898707, 0.827982, 0.853598, 0.768647, 0.848835, 0.783608, 0.838254, 0.757888, 0.824246, 0.754568, 0.808736, 0.737633, 0.797721, 0.731696, 0.789034, 0.721397, 0.780401, 0.716702, 0.771382, 0.70731, 0.764271, 0.703124

Ratios a(n)/A095326(n) converge as: 1, 1, 1, 1, 1, 0.923077, 0.95, 0.928571, 1.030769, 1, 1.039409, 1.012136, 1.005571, 0.973815, 0.97816, 0.997325, 1.018993, 1.00808, 1.009864, 1.002794,1.003497, 1.000397, 1.005197, 1.000665, 1.001903, 1.003022, 1.004856,1.00371, 1.004471, 1.001864, 1.001392, 1.001771, 1.002428

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)-A095294(n). Cf. A095052, A095053.

cp's OEIS Frontend

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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A095072 Primes in whose binary expansion the number of 0-bits is one more than the number of 1-bits.

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A095294 Number of A095284-primes in range ]2^n,2^(n+1)].

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A095295 Number of A095285-primes in range ]2^n,2^(n+1)].

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Author

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