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A337532 a(n) = Sum_{d1|n, d2|n, d1<=d2} [(d1 mod 2) = (d2 mod 2)], where [ ] is the Iverson bracket.

%I A337532 #16 Dec 21 2024 22:31:27
%S A337532 1,2,3,4,3,6,3,7,6,6,3,13,3,6,10,11,3,12,3,13,10,6,3,24,6,6,10,13,3,
%T A337532 20,3,16,10,6,10,27,3,6,10,24,3,20,3,13,21,6,3,39,6,12,10,13,3,20,10,
%U A337532 24,10,6,3,46,3,6,21,22,10,20,3,13,10,20,3,51,3,6,21,13,10,20,3,39
%N A337532 a(n) = Sum_{d1|n, d2|n, d1<=d2} [(d1 mod 2) = (d2 mod 2)], where [ ] is the Iverson bracket.
%C A337532 Number of distinct rectangles that can be made whose side lengths are divisors of n and whose length and width are either both odd or both even.
%H A337532 Robert Israel, <a href="/A337532/b337532.txt">Table of n, a(n) for n = 1..10000</a>
%F A337532 a(n) = A000217(A000005(A000265(n))) + A000217(A000005(A000265(n))*A007814(n)). - _Robert Israel_, Nov 01 2020
%p A337532 f:= proc(n) local t,m, n1,n2; t:= padic:-ordp(n,2);
%p A337532   m:= n/2^t;
%p A337532   n1:= numtheory:-tau(m);
%p A337532   n2:= n1*t;
%p A337532   (n1*(n1+1)+n2*(n2+1))/2;
%p A337532 end proc:
%p A337532 map(f, [$1..100]); # _Robert Israel_, Nov 01 2020
%t A337532 Table[Sum[Sum[KroneckerDelta[Mod[i, 2], Mod[k, 2]]*(1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]
%Y A337532 Cf. A000005, A000217, A000265, A007814.
%K A337532 nonn,look
%O A337532 1,2
%A A337532 _Wesley Ivan Hurt_, Aug 30 2020