A337363 a(n) = Sum_{d1|n, d2|n, d1

0, 0, 1, 2, 1, 4, 1, 5, 3, 5, 1, 12, 1, 5, 6, 9, 1, 13, 1, 13, 6, 5, 1, 25, 3, 5, 6, 14, 1, 25, 1, 14, 6, 5, 6, 33, 1, 5, 6, 26, 1, 25, 1, 14, 15, 5, 1, 42, 3, 14, 6, 14, 1, 26, 6, 26, 6, 5, 1, 61, 1, 5, 15, 20, 6, 26, 1, 14, 6, 27, 1, 62, 1, 5, 15, 14, 6, 26, 1, 43, 10, 5, 1, 62

Offset: 1

Author

Wesley Ivan Hurt, Aug 24 2020

Keywords

• nonn

Comments

Number of pairs of divisors of n, (d1,d2), with d1 < d2 such that d1 and d2 are nonconsecutive integers. For example, the 4 pairs for a(6) are (1,3), (1,6), (2,6) and (3,6).

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

Links

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

Crossrefs

Cf. A000005, A337362, A066446, 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 - 1}], {k, n}], {n, 100}]
Table[Count[Subsets[Divisors[n],{2}],?(#[[2]]-#[[1]]>1&)],{n,90}] (* _Harvey P. Dale, Mar 11 2023 *)

PARI

a(n) = sumdiv(n, d1, sumdiv(n, d2, (d1Michel Marcus, Aug 25 2020

Formula

a(n) = A337362(n) - A000005(n).

a(n) = A066446(n) - A129308(n). - Ridouane Oudra, Apr 16 2023