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

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, 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, 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

Offset: 1

Author

Wesley Ivan Hurt, Aug 30 2020

Keywords

• nonn
• look

Comments

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.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000217, A000265, A007814.

Programs

Maple

f:= proc(n) local t,m, n1,n2; t:= padic:-ordp(n,2);
  m:= n/2^t;
  n1:= numtheory:-tau(m);
  n2:= n1*t;
  (n1*(n1+1)+n2*(n2+1))/2;
end proc:
map(f, [$1..100]); # Robert Israel, Nov 01 2020

Mathematica

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}]

Formula

a(n) = A000217(A000005(A000265(n))) + A000217(A000005(A000265(n))*A007814(n)). - Robert Israel, Nov 01 2020