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A129308 a(n) = number of positive integers k such that k(k+1) divides n. 25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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

LINKS

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

P. Erdos 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)/2)/(1-x^(n*(n+1)/2))). - Joerg Arndt, Jan 30 2011

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

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.

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

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

Cf. A000005, A003601, A007862, A027750, A033676, A060680, A060681, A072627, A181063, A199970, A328026, A328165, A328166.

Sequence in context: A262202 A284413 A323879 * A159200 A318721 A219201

Adjacent sequences:  A129305 A129306 A129307 * A129309 A129310 A129311

KEYWORD

nonn

AUTHOR

Leroy Quet, May 26 2007

EXTENSIONS

More terms from Stefan Steinerberger, May 27 2007

Extended by Ray Chandler, Jun 24 2008

STATUS

approved

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Last modified November 22 06:26 EST 2019. Contains 329389 sequences. (Running on oeis4.)