A036349 Numbers whose sum of prime factors (taken with multiplicity) is even.

1, 2, 4, 8, 9, 15, 16, 18, 21, 25, 30, 32, 33, 35, 36, 39, 42, 49, 50, 51, 55, 57, 60, 64, 65, 66, 69, 70, 72, 77, 78, 81, 84, 85, 87, 91, 93, 95, 98, 100, 102, 110, 111, 114, 115, 119, 120, 121, 123, 128, 129, 130, 132, 133, 135, 138, 140, 141, 143, 144, 145, 154, 155

Offset: 1

Author

Patrick De Geest, Dec 15 1998

Keywords

• nonn

Comments

A multiplicative semigroup; if m and n are in the sequence then so is m*n. - David James Sycamore, Jul 17 2018

From Peter Munn, Jul 19 2020: (Start)

Also closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897.

A number is listed if and only if it has an even number of odd prime factors, counting repetitions; equivalently, if and only if it is the product of a term of A046337 and a power of 2 (term of A000079).

(End)

Examples

			141 = 3 * 47 is a term since the sum 3 + 47 = 50 is even.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001414 (sopfr), A059897.

Complement of A335657.

Sequences with similar definitions: A036350, A046363, A289142.

Subsequences: A000079, A028982, A046337, A056913.

Programs

Maple

filter:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2])::even end proc:
select(filter, [$1..200]); # Robert Israel, Jul 15 2020

Mathematica

Select[Range[160],EvenQ[Total[Times@@@FactorInteger[#]]]&] (* Harvey P. Dale, Sep 21 2011 *)

PARI

isok(n) = my(f=factor(n)); (sum(k=1, #f~, f[k,1]*f[k,2]) % 2) == 0; \\ Michel Marcus, Jul 19 2018

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

Extensions

First term (2) from Harvey P. Dale, Sep 21 2011

First term (1) from David James Sycamore, Jul 17 2018