%I #16 Feb 04 2023 17:22:39

%S 1,1,0,1,0,1,0,1,0,0,0,2,0,0,0,1,0,1,0,1,0,0,0,5,0,0,0,1,0,3,0,1,0,0,

%T 0,5,0,0,0,3,0,2,0,0,0,0,0,10,0,0,0,0,0,1,0,2,0,0,0,31,0,0,0,1,0,1,0,

%U 0,0,1,0,26,0,0,0,0,0,1,0,6,0,0,0,23,0,0,0,1,0,20,0,0,0,0,0,21,0,0,0,1

%N Number of ways to write n as sum of distinct divisors of n+1.

%C a(A085492(n)) = 0; a(A085493(n)) > 0; a(A085494(n)) = 1.

%H Alois P. Heinz, <a href="/A085491/b085491.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = [x^n] Product_{d divides (n+1)} (1 + x^d). - _Alois P. Heinz_, Feb 04 2023

%e n=11, divisors of 12=11+1 that are not greater 11: {1,2,3,4,6}, 11=6+5=6+4+1, therefore a(11)=2.

%p a:= proc(m) option remember; local b, l; b, l:=

%p proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

%p b(n, i-1)+`if`(l[i]>n, 0, b(n-l[i], i-1))))

%p end, sort([numtheory[divisors](m+1)[]]);

%p forget(b); b(m, nops(l)-1)

%p end:

%p seq(a(n), n=0..120); # _Alois P. Heinz_, Mar 12 2019

%t a[n_] := Module[{dd}, dd = Select[Divisors[n+1], # <= n&]; Select[ IntegerPartitions[n, dd // Length, dd], Reverse[#] == Union[#]&] // Length]; Array[a, 100, 0] (* _Jean-François Alcover_, Mar 12 2019 *)

%Y Cf. A085496.

%K nonn

%O 0,12

%A _Reinhard Zumkeller_, Jul 03 2003

%E a(0)=1 prepended by _Alois P. Heinz_, Mar 12 2019