A337363 a(n) = Sum_{d1|n, d2|n, d1<d2} (1 - [d1 + 1 = d2]), where [ ] is the Iverson bracket.

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

1,4

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

Formula

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

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

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 *)

Prog

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

Crossrefs

Cf. A000005, A337362, A066446, A129308.

Sequence in context: A006306 A322100 A277100 * A339243 A214579 A083711

Adjacent sequences: A337360 A337361 A337362 * A337364 A337365 A337366

Keyword

nonn

Author

Wesley Ivan Hurt, Aug 24 2020

Status

approved