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A210442 Number of partitions of n into proper divisors of n, cf. A027751.

%I A210442 #19 Oct 27 2023 19:54:31
%S A210442 1,0,1,1,3,1,7,1,9,4,10,1,44,1,13,13,35,1,80,1,91,17,19,1,457,6,22,22,
%T A210442 155,1,741,1,201,25,28,25,2233,1,31,29,1369,1,1653,1,336,285,37,1,
%U A210442 9675,8,406,37,453,1,3131,37,3064,41,46,1,73154,1,49,492,1827
%N A210442 Number of partitions of n into proper divisors of n, cf. A027751.
%C A210442 For n > 0: a(A000040(n)) = 1 and a(A002808(n)) > 1.
%H A210442 Alois P. Heinz, <a href="/A210442/b210442.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 0..197 from Reinhard Zumkeller)
%p A210442 with(numtheory):
%p A210442 a:= proc(n) local b, l; l:= sort([(divisors(n) minus {n})[]]):
%p A210442       b:= proc(m, i) option remember; `if`(m=0 or i=1, 1,
%p A210442             `if`(i<1, 0, b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
%p A210442           end; forget(b):
%p A210442       b(n, nops(l))
%p A210442     end:
%p A210442 seq(a(n), n=0..100); # _Alois P. Heinz_, Jan 29 2013
%t A210442 a[n_] := Module[{b, l}, l = Most[Divisors[n]]; b[m_, i_] := b[m, i] = If[m==0 || i==1, 1, If[i<1, 0, b[m, i-1] + If[l[[i]]>m, 0, b[m-l[[i]], i]]]]; b[n, Length[l]]]; a[0]=1; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 02 2017, after _Alois P. Heinz_ *)
%o A210442 (Haskell)
%o A210442 a210442 n = p (a027751_row n) n where
%o A210442    p _          0 = 1
%o A210442    p []         _ = 0
%o A210442    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%Y A210442 Cf. A065205, A211110, A018818.
%K A210442 nonn,look
%O A210442 0,5
%A A210442 _Reinhard Zumkeller_, Jan 21 2013