A337532 - OEIS

%I #13 Nov 02 2020 11:52:17

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

%C 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 Robert Israel, <a href="/A337532/b337532.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A000217(A000005(A000265(n)) + A000217(A000005(A000265(n))*A007814(n)). - _Robert Israel_, Nov 01 2020

%p f:= proc(n) local t,m, n1,n2; t:= padic:-ordp(n,2);

%p   m:= n/2^t;

%p   n1:= numtheory:-tau(m);

%p   n2:= n1*t;

%p   (n1*(n1+1)+n2*(n2+1))/2;

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Nov 01 2020

%t 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 Cf. A000005, A000217, A000265, A007814.

%K nonn,look

%O 1,2

%A _Wesley Ivan Hurt_, Aug 30 2020