A129308 - OEIS

A129308

a(n) is the number of positive integers k such that k*(k+1) divides n.

0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 5, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 4, 0, 1, 0, 1, 0, 4, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 1, 0

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OFFSET

1,6

COMMENTS

The usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.

In other words, a(n) is the number of oblong numbers (A002378) dividing n. - Bernard Schott, Jul 29 2022

LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000

P. Erdős and R. R. Hall, On some unconventional problems on the divisors of integers, J. Austral. Math. Soc., Ser. A, 25, 479-485 (1978).

MathOverflow, On the number of consecutive divisors of an integer.

FORMULA

a(2n-1) = 0; a(2n) = A007862(n). - Ray Chandler, Jun 24 2008

G.f.: Sum_{n>=1} x^(n*(n+1))/(1-x^(n*(n+1))). - Joerg Arndt, Jan 30 2011 [modified by Ilya Gutkovskiy, Apr 14 2021]

a(n) = A000005(n) - A137921(n), where A137921(n) is the number of maximal runs of successive divisors of n. - Gus Wiseman, Oct 15 2019

a(n) = Sum_{d|n} A005369(d). - Ridouane Oudra, Jan 22 2021

a(n) = A195155(n)-1. - Antti Karttunen, Feb 21 2023

From Amiram Eldar, Dec 31 2023: (Start)

a(n) = A088722(n) + A059841(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. (End)

EXAMPLE

The divisors of 20 are 1,2,4,5,10,20. Of these there are two that are of the form k(k+1): 2 = 1*2 and 20 = 4*5. So a(2) = 2.

MATHEMATICA

a = {}; For[n = 1, n < 90, n++, k = 1; co = 0; While[k < Sqrt[n], If[IntegerQ[ n/(k*(k + 1))], co++ ]; k++ ]; AppendTo[a, co]]; a (* Stefan Steinerberger, May 27 2007 *)

Table[Count[Differences[Divisors[n]], 1], {n, 30}] (* Gus Wiseman, Oct 15 2019 *)

PROG

(PARI) a(n)=sumdiv(n, d, n%(d+1)==0); \\ Michel Marcus, Jan 06 2015

CROSSREFS

Positions of 0's and 1's are A088725, whose characteristic function is A360128.

First appearance of n is A287142(n), with sorted version A328450.

The longest run of divisors of n has length A055874(n).

One less than A195155.