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A211110 Number of partitions of n into divisors > 1 of n.

%I A211110 #19 Jun 30 2015 06:16:01
%S A211110 1,0,1,1,2,1,3,1,4,2,3,1,12,1,3,3,10,1,15,1,16,3,3,1,80,2,3,5,20,1,94,
%T A211110 1,36,3,3,3,280,1,3,3,158,1,154,1,28,25,3,1,1076,2,29,3,32,1,255,3,
%U A211110 262,3,3,1,7026,1,3,32,202,3,321,1,40,3,302,1,12072,1
%N A211110 Number of partitions of n into divisors > 1 of n.
%C A211110 a(A000040(n)) = 1; a(A002808(n)) > 1;
%C A211110 a(A001248(n)) = 2; a(A080257(n)) > 2;
%C A211110 a(A006881(n)) = 3; a(A033942(n)) > 3.
%H A211110 Alois P. Heinz, <a href="/A211110/b211110.txt">Table of n, a(n) for n = 0..10000</a>
%e A211110 a(10) = #{10, 5+5, 2+2+2+2+2} = 3;
%e A211110 a(11) = #{11} = 1;
%e A211110 a(12) = #{12, 6+6, 6+4+2, 6+3+3, 6+2+2+2, 4+4+4, 4+4+2+2, 4+3+3+2, 4+2+2+2+2, 3+3+3+3, 3+3+2+2+2, 6x2} = 12;
%e A211110 a(13) = #{13} = 1;
%e A211110 a(14) = #{14, 7+7, 2+2+2+2+2+2+2} = 3;
%e A211110 a(15) = #{15, 5+5+5, 3+3+3+3+3} = 3.
%p A211110 with(numtheory):
%p A211110 a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1})[]]):
%p A211110       b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
%p A211110              b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
%p A211110           end; forget(b):
%p A211110       b(n, nops(l))
%p A211110     end:
%p A211110 seq(a(n), n=0..100); # _Alois P. Heinz_, Feb 05 2014
%t A211110 a[n_] := Module[{b, l}, l = Rest[Divisors[n]]; b[m_, i_] := b[m, i] = If[m==0, 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_, Jun 30 2015, after _Alois P. Heinz_ *)
%o A211110 (Haskell)
%o A211110 a211110 n = p (tail $ a027750_row n) n where
%o A211110    p _      0 = 1
%o A211110    p []     _ = 0
%o A211110    p ks'@(k:ks) m | m < k     = 0
%o A211110                   | otherwise = p ks' (m - k) + p ks m
%o A211110 (PARI) isokp(p, n) = {for (k=1, #p, if ((p[k]==1) || (n % p[k]), return (0));); return (1);}
%o A211110 a(n) = {my(nb = 0); forpart(p=n, if (isokp(p,n), nb++)); nb;} \\ _Michel Marcus_, Jun 30 2015
%Y A211110 Cf. A211111, A018818, A027750.
%Y A211110 Cf. A210442.
%K A211110 nonn
%O A211110 0,5
%A A211110 _Reinhard Zumkeller_, Apr 01 2012