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

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, 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, 16, 8, 4, 48, 4, 8, 36, 7, 16, 32, 4, 12, 16, 32, 4, 36, 4, 8, 36, 12, 16, 32, 4, 20

Offset

1,2

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

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

a(n) = tau(n)*(tau(2*n) - tau(n)), with tau(n) = A000005(n). - Ridouane Oudra, Feb 19 2023

Example

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.

Maple

with(numtheory): seq(tau(n)*(tau(2*n)-tau(n)), n=1..100); # Ridouane Oudra, Feb 19 2023

Mathematica

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}]

Prog

(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

Crossrefs

Cf. A337532, A000005.

Sequence in context: A087794 A050514 A229047 * A133702 A328486 A349606

Adjacent sequences: A335838 A335839 A335840 * A335842 A335843 A335844

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Sep 13 2020

Status

approved