A088722 Number of divisors d>1 of n such that d+1 also divides n.

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

Offset

1,12

Comments

Also, number of partitions of n into two distinct parts (s,t), s<t, such that s|n and t|s*n. - Wesley Ivan Hurt, Jan 16 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Formula

a(A088723(n)) > 0, a(A088724(n)) = 1, a(A088725(n)) = 0.

a(A088726(n)) = n, a(k) <> n, for n < A088726(n).

a(2n+1) = 0. - Ray Chandler, May 29 2008

a(n) = Sum_{d|n, (d+1)|n, d>1} 1. - Wesley Ivan Hurt, Jan 16 2022

From Amiram Eldar, Dec 31 2023: (Start)

a(n) = A129308(n) - A059841(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. (End)

Example

n=144: divisors(144) = {1,2,3,4,6,8,9,12,16,18,24,36,48,72,144}, there are a(144) = 3 divisors d>1 such that also d+1 divides 144: (2,3), (3,4) and (8,9).

Mathematica

Table[DivisorSum[n, 1 &, And[# > 1, Divisible[n, # + 1]] &], {n, 105}] (* Michael De Vlieger, Jul 12 2017 *)

Prog

(PARI) A088722(n) = sumdiv(n, d, (d>1)&&!(n%(d+1))); \\ Antti Karttunen, Jul 12 2017
(PARI) first(n) = my(v = vector(n), k); for(i=2, sqrtint(n), k=i*(i+1); for(j=1, n\k, v[j*k]++); v \\ David A. Corneth, Jul 12 2017

Crossrefs

Cf. A059841, A088723, A088724, A088725, A088726, A129308.

Sequence in context: A101638 A321379 A070141 * A336917 A122180 A363742

Adjacent sequences: A088719 A088720 A088721 * A088723 A088724 A088725

Keyword

nonn

Author

Reinhard Zumkeller, Oct 12 2003

Extensions

Extended by Ray Chandler, May 29 2008

Status

approved