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

%I A088722 #27 Aug 25 2024 02:44:52
%S A088722 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,
%T A088722 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,
%U A088722 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
%N A088722 Number of divisors d>1 of n such that d+1 also divides n.
%C A088722 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
%H A088722 Antti Karttunen, <a href="/A088722/b088722.txt">Table of n, a(n) for n = 1..10000</a>
%F A088722 a(A088723(n)) > 0, a(A088724(n)) = 1, a(A088725(n)) = 0.
%F A088722 a(A088726(n)) = n, a(k) <> n, for n < A088726(n).
%F A088722 a(2n+1) = 0. - _Ray Chandler_, May 29 2008
%F A088722 a(n) = Sum_{d|n, (d+1)|n, d>1} 1. - _Wesley Ivan Hurt_, Jan 16 2022
%F A088722 From _Amiram Eldar_, Dec 31 2023: (Start)
%F A088722 a(n) = A129308(n) - A059841(n).
%F A088722 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. (End)
%e A088722 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).
%t A088722 Table[DivisorSum[n, 1 &, And[# > 1, Divisible[n, # + 1]] &], {n, 105}] (* _Michael De Vlieger_, Jul 12 2017 *)
%o A088722 (PARI) A088722(n) = sumdiv(n,d,(d>1)&&!(n%(d+1))); \\ _Antti Karttunen_, Jul 12 2017
%o A088722 (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
%Y A088722 Cf. A059841, A088723, A088724, A088725, A088726, A129308.
%K A088722 nonn,easy
%O A088722 1,12
%A A088722 _Reinhard Zumkeller_, Oct 12 2003
%E A088722 Extended by _Ray Chandler_, May 29 2008