A081092 - OEIS

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

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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

A049084(A000120(a(n))) > 0; A081091, A000215 and A081093 are subsequences.

LINKS

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

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

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