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A088722
Number of divisors d>1 of n such that d+1 also divides n.
12
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
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
KEYWORD
nonn,easy
AUTHOR
Reinhard Zumkeller, Oct 12 2003
EXTENSIONS
Extended by Ray Chandler, May 29 2008
STATUS
approved