cp's OEIS Frontend

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

%I A036349 #43 Nov 02 2020 20:38:15
%S A036349 1,2,4,8,9,15,16,18,21,25,30,32,33,35,36,39,42,49,50,51,55,57,60,64,
%T A036349 65,66,69,70,72,77,78,81,84,85,87,91,93,95,98,100,102,110,111,114,115,
%U A036349 119,120,121,123,128,129,130,132,133,135,138,140,141,143,144,145,154,155
%N A036349 Numbers whose sum of prime factors (taken with multiplicity) is even.
%C A036349 A multiplicative semigroup; if m and n are in the sequence then so is m*n. - _David James Sycamore_, Jul 17 2018
%C A036349 From _Peter Munn_, Jul 19 2020: (Start)
%C A036349 Also closed under the commutative binary operation A059897(.,.), forming a subgroup of the positive integers under A059897.
%C A036349 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).
%C A036349 (End)
%H A036349 Robert Israel, <a href="/A036349/b036349.txt">Table of n, a(n) for n = 1..10000</a>
%F A036349 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
%e A036349 141 = 3 * 47 is a term since the sum 3 + 47 = 50 is even.
%p A036349 filter:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2])::even end proc:
%p A036349 select(filter, [$1..200]); # _Robert Israel_, Jul 15 2020
%t A036349 Select[Range[160],EvenQ[Total[Times@@@FactorInteger[#]]]&] (* _Harvey P. Dale_, Sep 21 2011 *)
%o A036349 (PARI) isok(n) = my(f=factor(n)); (sum(k=1, #f~, f[k,1]*f[k,2]) % 2) == 0; \\ _Michel Marcus_, Jul 19 2018
%Y A036349 Cf. A001414 (sopfr), A059897.
%Y A036349 Complement of A335657.
%Y A036349 Sequences with similar definitions: A036350, A046363, A289142.
%Y A036349 Subsequences: A000079, A028982, A046337, A056913.
%K A036349 nonn
%O A036349 1,2
%A A036349 _Patrick De Geest_, Dec 15 1998
%E A036349 First term (2) from _Harvey P. Dale_, Sep 21 2011
%E A036349 First term (1) from _David James Sycamore_, Jul 17 2018