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A335841 Number of distinct rectangles that can be made with one even and one odd side length that are divisors of 2n.

%I A335841 #20 Feb 19 2023 12:17:49
%S A335841 1,2,4,3,4,8,4,4,9,8,4,12,4,8,16,5,4,18,4,12,16,8,4,16,9,8,16,12,4,32,
%T A335841 4,6,16,8,16,27,4,8,16,16,4,32,4,12,36,8,4,20,9,18,16,12,4,32,16,16,
%U A335841 16,8,4,48,4,8,36,7,16,32,4,12,16,32,4,36,4,8,36,12,16,32,4,20
%N A335841 Number of distinct rectangles that can be made with one even and one odd side length that are divisors of 2n.
%C A335841 If p > 2 is prime, a(p) = 4. There are 4 rectangles that can be made with one even and one odd side length that are divisors of 2p: 1 X 2, 1 X 2p, 2 X p, and p X 2p. - _Wesley Ivan Hurt_, May 21 2021
%H A335841 Antti Karttunen, <a href="/A335841/b335841.txt">Table of n, a(n) for n = 1..20000</a>
%F A335841 a(n) = Sum_{d1|(2*n), d2|(2*n), d1<d2} (1 - [d1 mod 2 = d2 mod 2]), where [ ] is the Iverson bracket.
%F A335841 a(n) = tau(n)*(tau(2*n) - tau(n)), with tau(n) = A000005(n). - _Ridouane Oudra_, Feb 19 2023
%e A335841 a(6) = 8; The divisors of 2*6 = 12 are {1,2,3,4,6,12}. There are 8 distinct rectangles with one odd and one even side length using these divisors. They are 1 X 2, 1 X 4, 1 X 6, 1 X 12, 2 X 3, 3 X 4, 3 X 6, and 3 X 12.
%p A335841 with(numtheory): seq(tau(n)*(tau(2*n)-tau(n)), n=1..100); # _Ridouane Oudra_, Feb 19 2023
%t A335841 Table[Sum[Sum[KroneckerDelta[Mod[i + 1, 2], Mod[k, 2]]*(1 - Ceiling[2n/k] + Floor[2n/k]) (1 - Ceiling[2n/i] + Floor[2n/i]), {i, k-1}], {k, 2n}], {n, 100}]
%o A335841 (PARI) A335841(n) = { my(ds=divisors(2*n)); sum(i=2, #ds, sum(j=1, i-1, (ds[i]+ds[j])%2)); }; \\ _Antti Karttunen_, Dec 09 2021
%Y A335841 Cf. A337532, A000005.
%K A335841 nonn,easy
%O A335841 1,2
%A A335841 _Wesley Ivan Hurt_, Sep 13 2020