A081092 Primes having a prime number of 1's in their binary representation.

3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443

Offset

1,1

Comments

Same as primes with prime binary digit sum.

Primes with prime decimal digit sum are A046704.

Sum_{a(n) < x} 1/a(n) is asymptotic to log(log(log(x))) as x -> infinity; see Harman (2012). Thus the sequence is infinite. - Jonathan Sondow, Jun 09 2012

A081091, A019434 and A081093 are subsequences.

Includes all the Mersenne primes (A000668). - Amiram Eldar, Feb 01 2026

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Glyn Harman, Counting Primes whose Sum of Digits is Prime, J. Integer Seq., 15 (2012), Article 12.2.2.

Formula

A049084(A000120(a(n))) > 0.

Example

15th prime = 47 = '101111' with five 1's, therefore 47 is in the sequence.

Maple

q:= n-> isprime(n) and isprime(add(i, i=Bits[Split](n))):
select(q, [$1..500])[];  # Alois P. Heinz, Sep 28 2023

Mathematica

Clear[BinSumOddQ]; BinSumPrimeQ[a_]:=Module[{i, s=0}, s=0; For[i=1, i<=Length[IntegerDigits[a, 2]], s+=Extract[IntegerDigits[a, 2], i]; i++ ]; PrimeQ[s]]; lst={}; Do[p=Prime[n]; If[BinSumPrimeQ[p], AppendTo[lst, p]], {n, 4!}]; lst (* Vladimir Joseph Stephan Orlovsky, Apr 06 2009 *)
Select[Prime[Range[100]], PrimeQ[Apply[Plus, IntegerDigits[#, 2]]] &] (* Jonathan Sondow, Jun 09 2012 *)

Prog

(Haskell)
a081092 n = a081092_list !! (n-1)
a081092_list = filter ((== 1) . a010051') a052294_list
-- Reinhard Zumkeller, Nov 16 2012
(PARI) lista(nn) = {forprime(p=2, nn, if (isprime(hammingweight(p)), print1(p, ", ")); ); } \\ Michel Marcus, Jan 16 2015
(Python)
from sympy import isprime
def ok(n): return isprime(n.bit_count()) and isprime(n)
print([k for k in range(444) if ok(k)]) # Michael S. Branicky, Dec 27 2023

Crossrefs

Intersection of A000040 and A052294.

Cf. A000120, A000668, A019434, A046704, A049084, A081091, A081093.

Sequence in context: A386337 A155026 A295705 * A291360 A269326 A163422

Adjacent sequences: A081089 A081090 A081091 * A081093 A081094 A081095

Keyword

nonn,base,easy

Author

Reinhard Zumkeller, Mar 05 2003

Status

approved