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

1,1

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.

Links

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

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Positions of odd numbers in A001414.

Complement of A036349.

Cf. A000040.

Sequence in context: A232682 A364289 A352826 * A038550 A204232 A028730

Adjacent sequences: A335654 A335655 A335656 * A335658 A335659 A335660

Keyword

nonn

Author

Antti Karttunen and Peter Munn, Jul 09 2020

Status

approved