A095077 Primes with four 1-bits in their binary expansion.

23, 29, 43, 53, 71, 83, 89, 101, 113, 139, 149, 163, 197, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 523, 547, 593, 643, 673, 773, 1031, 1049, 1061, 1091, 1093, 1097, 1217, 1283, 1289, 1297, 1409, 1553, 1601, 2069, 2083, 2089, 2129

Offset: 1

Author

Antti Karttunen, Jun 01 2004

Keywords

• nonn
• easy
• base

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Crossrefs

Subsequence of A027699. First differs from A085448 at n = 19, where a(n)=337, while A085448 continues from there with 311, whose binary expansion has six 1-bits, not four. Cf. A095057.

Cf. A000215 (primes having two bits set), A081091 (three bits set).

Cf. A264908.

Programs

Mathematica

Select[Prime[Range[320]], Plus@@IntegerDigits[#, 2] == 4 &] (* Alonso del Arte, Jan 11 2011 *)
Select[ Flatten[ Table[2^i + 2^j + 2^k + 1, {i, 3, 11}, {j, 2, i - 1}, {k, j - 1}]], PrimeQ] (* Robert G. Wilson v, Jul 30 2016 *)

PARI

bits1_4(x) = { nB = floor(log(x)/log(2)); z = 0;
for(i=0,nB,if(bittest(x,i),z++;if(z>4,return(0););););
if(z == 4, return(1);, return(0););};
forprime(x=17,2129,if(bits1_4(x),print1(x, ", ");););
\\ Washington Bomfim, Jan 11 2011

PARI

is(n)=isprime(n) && hammingweight(n)==4 \\ Charles R Greathouse IV, Jul 30 2016

PARI

list(lim)=my(v=List(),t); for(a=3,logint(lim\=1,2), for(b=2,a-1, for(c=1,b-1, t=1<lim, return(Vec(v))); if(isprime(t), listput(v,t))))); Vec(v) \\ Charles R Greathouse IV, Jul 30 2016

Python

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A095077_gen(): # generator of terms
    return filter(isprime,map(lambda s:int('1'+''.join(s)+'1',2),(s for l in count(2) for s in multiset_permutations('0'*(l-2)+'11'))))
A095077_list = list(islice(A095077_gen(),30)) # Chai Wah Wu, Jul 19 2022