

A335657


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


4



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
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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: A154663 A028983 A232682 * A038550 A204232 A028730
Adjacent sequences: A335654 A335655 A335656 * A335658 A335659 A335660


KEYWORD

nonn


AUTHOR

Antti Karttunen and Peter Munn, Jul 09 2020


STATUS

approved



