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A357311 Number of partitions of n into divisors of n that are smaller than sqrt(n).

%I A357311 #10 Sep 24 2022 13:45:44
%S A357311 1,0,1,1,1,1,4,1,5,1,6,1,19,1,8,6,9,1,37,1,36,8,12,1,169,1,14,10,64,1,
%T A357311 247,1,81,12,18,8,478,1,20,14,405,1,512,1,144,82,24,1,2825,1,146,18,
%U A357311 196,1,1000,12,743,20,30,1,19858,1,32,112,289,14,1728,1,324,24,1105
%N A357311 Number of partitions of n into divisors of n that are smaller than sqrt(n).
%H A357311 Alois P. Heinz, <a href="/A357311/b357311.txt">Table of n, a(n) for n = 0..10000</a>
%F A357311 a(n) = [x^n] Product_{d|n, d < sqrt(n)} 1 / (1 - x^d).
%p A357311 a:= proc(n) option remember; uses numtheory; local b, l;
%p A357311       l:= sort([select(x-> is(x<sqrt(n)), divisors(n))[]]):
%p A357311       b:= proc(m, i) option remember; `if`(m=0 or i=1, 1,
%p A357311             `if`(i<1, 0, b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
%p A357311           end; forget(b):
%p A357311       b(n, nops(l))
%p A357311     end:
%p A357311 seq(a(n), n=0..70);  # _Alois P. Heinz_, Sep 23 2022
%t A357311 a[n_] := SeriesCoefficient[Product[1/(1 - Boole[d < Sqrt[n]] x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 70}]
%Y A357311 Cf. A018818, A210442, A293813, A327642, A357312.
%K A357311 nonn
%O A357311 0,7
%A A357311 _Ilya Gutkovskiy_, Sep 23 2022