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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. 0

%I

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

%C 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

%F a(n) = Sum_{d1|(2*n), d2|(2*n), d1<d2} (1 - [d1 mod 2 = d2 mod 2]), where [ ] is the Iverson bracket.

%e 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.

%t 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}]

%Y Cf. A337532.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Sep 13 2020

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Last modified June 23 12:08 EDT 2021. Contains 345401 sequences. (Running on oeis4.)