A081093 -id:A081093 - cp's OEIS frontend

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

Reinhard Zumkeller, Mar 05 2003

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.

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

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.

Cf. A000040, A000120, A046704, A081093.

Subsequence of A052294.

a081092 n = a081092_list !! (n-1)
a081092_list = filter ((== 1) . a010051') a052294_list
-- Reinhard Zumkeller, Nov 16 2012

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

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 *)

lista(nn) = {forprime(p=2, nn, if (isprime(hammingweight(p)), print1(p, ", ")););} \\ Michel Marcus, Jan 16 2015

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

A102029 Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.

Original entry on oeis.org

4, 6, 14, 15, 55, 95, 247, 447, 511, 1535, 2047, 7167, 12287, 32255, 49151, 98303, 196607, 393215, 983039, 1572863, 3145727, 6291455, 8388607, 33423359, 50331647, 117440511, 201326591, 528482303, 805306367, 1879048191, 3221225471

Offset: 1

Jonathan Vos Post, Jun 23 2007

Semiprime analog of A061712. Extended by Stefan Steinerberger. Includes the subset Mersenne semiprimes A092561.

			a(1) = 4 because the first semiprime A001358(1) is 4 (base 10) which is written 100 in binary, the latter representation having exactly 1 one.
a(2) = 6 since A001358(2) = 6 = 110 (base 2) has exactly 2 ones.
a(4) = 15 since A001358(6) = 15 = 1111 (base 2) has exactly 4 ones and, as it also has no zeros, is the smallest of the Mersenne semiprimes.

Donovan Johnson, Table of n, a(n) for n = 1..250

Cf. A000043, A000120, A000337, A000668, A001358, A007088, A061712, A085724, A089226, A089998, A089999, A091991, A092558, A092559, A092561, A092562, A081093, A102782, A110472, A110699, A110700.

Join[{4},Table[SelectFirst[Sort[FromDigits[#,2]&/@Permutations[ Join[ PadRight[{}, n,1],{0}]]],PrimeOmega[#]==2&],{n,2,40}]] (* Harvey P. Dale, Feb 06 2015 *)

cp's OEIS Frontend

A093535 Duplicate of A081093.

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

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A102029 Smallest semiprime with Hamming weight n (i.e., smallest semiprime with exactly n ones when written in binary), or -1 if no such number exists.

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