A337362 Number of pairs of divisors of n, (d1,d2), with d1 <= d2 such that d1 and d2 are nonconsecutive integers.

1, 2, 3, 5, 3, 8, 3, 9, 6, 9, 3, 18, 3, 9, 10, 14, 3, 19, 3, 19, 10, 9, 3, 33, 6, 9, 10, 20, 3, 33, 3, 20, 10, 9, 10, 42, 3, 9, 10, 34, 3, 33, 3, 20, 21, 9, 3, 52, 6, 20, 10, 20, 3, 34, 10, 34, 10, 9, 3, 73, 3, 9, 21, 27, 10, 34, 3, 20, 10, 35, 3, 74, 3, 9, 21, 20, 10, 34, 3, 53, 15

Offset: 1

Author

Wesley Ivan Hurt, Aug 24 2020

Keywords

• nonn

Comments

Number of distinct rectangles that can be made using the divisors of n as side lengths and whose length is never one more than its width.

Examples

			a(6) = 8; The divisors of 6 are {1,2,3,6}. There are 8 divisor pairs, (d1,d2), with d1 <= d2 that do not contain consecutive integers. They are (1,1), (1,3), (1,6), (2,2), (2,6), (3,3), (3,6) and (6,6). So a(6) = 8.
		

Links

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

Crossrefs

Cf. A000005, A337363.

Cf. also A335841, A337333, A184389, A129308.

Programs

Mathematica

Table[Sum[Sum[(1 - KroneckerDelta[i + 1, k]) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 100}]

PARI

a(n) = sumdiv(n, d1, sumdiv(n, d2, (d1<=d2) && (d1 + 1 != d2))); \\ Michel Marcus, Aug 25 2020

Formula

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

a(n) = A337363(n) + A000005(n).

a(n) = A184389(n) - A129308(n). - Ridouane Oudra, Apr 15 2023