A335657 Numbers whose prime factors (including repetitions) sum to an odd number.

3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 24, 26, 27, 28, 29, 31, 34, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 52, 53, 54, 56, 58, 59, 61, 62, 63, 67, 68, 71, 73, 74, 75, 76, 79, 80, 82, 83, 86, 88, 89, 90, 92, 94, 96, 97, 99, 101, 103, 104, 105, 106, 107, 108, 109, 112, 113, 116, 117, 118, 122, 124, 125

Offset: 1

Author

Antti Karttunen and Peter Munn, Jul 09 2020

Keywords

• nonn

Comments

Every positive integer, m, can be written uniquely as a product of primes (A000040). Rewrite with addition substituted for multiplication. m is in the sequence if and only if the result, which is A001414(m), is odd.

Examples

			12 = 2 * 2 * 3 (where the factors are prime numbers). Substituting addition for multiplication we get 2 + 2 + 3 = 7, which is odd. So 12 is in the sequence.
50 = 2 * 5 * 5. Substituting addition for multiplication we get 2 + 5 + 5 = 12, which is not odd. So 50 is not in the sequence.
1, written as a product of primes, is the empty product (1 has zero prime factors). Substituting addition for multiplication gives the empty sum, which evaluates as 0, which is even, not odd. So 1 is not in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Positions of odd numbers in A001414.

Complement of A036349.

Cf. A000040.

Programs

Mathematica

Select[Range[2, 125], OddQ[Plus @@ Times @@@ FactorInteger[#]] &] (* Amiram Eldar, Jul 11 2020 *)

PARI

isA335657(n) = (((n=factor(n))[, 1]~*n[, 2])%2); \\ After code in A001414.

Formula

Sum_{n>=1} 1/a(n)^s = (zeta(s) - ((2^s + 1)/(2^s - 1))*zeta(2*s)/zeta(s))/2 for Re(s)>1. - Amiram Eldar, Nov 02 2020