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A345452 Positive integers with an even number of prime factors (counting repetitions) that sum to an even number.

%I A345452 #18 Jun 25 2021 04:09:02
%S A345452 1,4,9,15,16,21,25,33,35,36,39,49,51,55,57,60,64,65,69,77,81,84,85,87,
%T A345452 91,93,95,100,111,115,119,121,123,129,132,133,135,140,141,143,144,145,
%U A345452 155,156,159,161,169,177,183,185,187,189,196,201,203,204,205,209,213,215
%N A345452 Positive integers with an even number of prime factors (counting repetitions) that sum to an even number.
%C A345452 Numbers with an even number of even prime factors and an even number of odd prime factors.
%C A345452 The representation (as defined in A206284) of polynomials with nonnegative integer coefficients that are in the ideal of the polynomial ring Z[x] generated by x^2+x and 2.
%C A345452 The above property arises because the sequence lists the integers in the multiplicative subgroup of positive rational numbers generated by the squares of primes (A001248) and the products of two consecutive odd primes (A006094\{6}).
%C A345452 The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 4 and 15. For example, A003961(4) = 9, A003961(9) = 25, A003961(15) = 35, 15 * 35 = 525, 525/25 = 21. Alternatively, the sequence may be defined as the closure of A046337 under multiplication by 4.
%C A345452 From the properties of subgroups of the positive rationals we know that if we take an absent positive integer m and divide all terms that are multiples of m by m, we get all the integers in the same subgroup coset as m, and we can expect some of the nice properties here to carry over to the resulting set. Specifically, dividing the even terms by 2 gives all numbers with an odd number of prime factors that sum to an even number; dividing all terms divisible by an odd prime p by p, gives all numbers with an odd number of prime factors that sum to an odd number. The positive integers satisfying the 4th of the 4 possibilities are generated similarly, dividing by 6 (for example).
%C A345452 Numbers whose squarefree part is in A056913.
%C A345452 Term by term, the sequence is one half of its complement within A036349.
%H A345452 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Group.html">Group</a>.
%H A345452 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polynomial_ring">Polynomial ring</a>.
%F A345452 {a(n) : n >= 1} = {m >= 1 : A001222(m) mod 2 = A001414(m) mod 2 = 0}.
%F A345452 {A036349(n) : n >= 1} = {a(n) : n >= 1} U {2 * a(n) : n >= 1}.
%F A345452 {A028260(n) : n >= 1} = {a(n) : n >= 1} U {A307150(a(n)) : n >= 1}.
%F A345452 For odd prime p, {A003159(n) : n >= 1} = {a(n) : n >= 1} U {A059897(a(n), p) : n >= 1}.
%e A345452 The definition specifies that we count repeated prime factors.
%e A345452 6 = 2 * 3; the sum of these prime factors is 2 + 3 = 5, an odd number; so 6 is not in the sequence.
%e A345452 50 = 2 * 5 * 5 has 3 prime factors and 3 is an odd number; so 50 is not in the sequence.
%e A345452 60 = 2 * 2 * 3 * 5 has 4 prime factors and 4 is an even number; the sum of these factors is 2 + 2 + 3 + 5 = 12, also an even number; so 60 is in the sequence.
%e A345452 1 has 0 prime factors, which sum to 0 (the empty sum). 0 is even, so 1 is in the sequence.
%t A345452 {1}~Join~Select[Range@1000,(s=Flatten[Table@@@FactorInteger[#]];And@@EvenQ@{Length@s,Total@s})&] (* _Giorgos Kalogeropoulos_, Jun 24 2021 *)
%o A345452 (PARI) iseven(x) = ((x%2) == 0);
%o A345452 isok(m) = my(f=factor(m)); iseven(sum(k=1, #f~, f[k,1]*f[k,2])) && iseven(sum(k=1, #f~, f[k,2])); \\ _Michel Marcus_, Jun 24 2021
%o A345452 (PARI) is(n) = bigomega(n)%2 == 0 && valuation(n, 2)%2 == 0 \\ _David A. Corneth_, Jun 24 2021
%o A345452 (Python)
%o A345452 from sympy import factorint
%o A345452 def ok(n):
%o A345452     f = factorint(n)
%o A345452     return sum(f.values())%2 == 0 and sum(p*f[p] for p in f)%2 == 0
%o A345452 print(list(filter(ok, range(1, 216)))) # _Michael S. Branicky_, Jun 24 2021
%Y A345452 Cf. A001222, A001414, A003961, A059897, A307150.
%Y A345452 Intersection of any 2 of A003159, A028260, A036349.
%Y A345452 Other lists that have conditions on the number of odd prime factors: A046337, A072978.
%Y A345452 Subsequences: A001248, A006094\{6}, A046315, A056913.
%K A345452 nonn,easy
%O A345452 1,2
%A A345452 _Peter Munn_, Jun 20 2021