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

Author

Reinhard Zumkeller, Oct 12 2003

Keywords

• nonn
• easy

Comments

Also, number of partitions of n into two distinct parts (s,t), sWesley Ivan Hurt, Jan 16 2022

Examples

			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).
		

Links

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

Crossrefs

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

Programs

Mathematica

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

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

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)

Extensions

Extended by Ray Chandler, May 29 2008