A337362 - OEIS

The OEIS is supported by the many generous donors to the OEIS Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

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

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

EXAMPLE

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.

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}]

PROG

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

CROSSREFS

Cf. A000005, A337363.

Cf. also A335841, A337333, A184389, A129308.

Sequence in context: A346091 A107685 A161984 * A125677 A051358 A346248

Adjacent sequences: A337359 A337360 A337361 * A337363 A337364 A337365

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Aug 24 2020

STATUS

approved