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

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, 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, 262, 3, 3, 1, 7026, 1, 3, 32, 202, 3, 321, 1, 40, 3, 302, 1, 12072, 1

Offset: 0

Author

Reinhard Zumkeller, Apr 01 2012

Keywords

• nonn

Comments

a(A000040(n)) = 1; a(A002808(n)) > 1;

a(A001248(n)) = 2; a(A080257(n)) > 2;

a(A006881(n)) = 3; a(A033942(n)) > 3.

Examples

			a(10) = #{10, 5+5, 2+2+2+2+2} = 3;
a(11) = #{11} = 1;
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;
a(13) = #{13} = 1;
a(14) = #{14, 7+7, 2+2+2+2+2+2+2} = 3;
a(15) = #{15, 5+5+5, 3+3+3+3+3} = 3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A211111, A018818, A027750.

Cf. A210442.

Programs

Haskell

a211110 n = p (tail $ a027750_row n) n where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m

Maple

with(numtheory):
a:= proc(n) local b, l; l:= sort([(divisors(n) minus {1})[]]):
      b:= proc(m, i) option remember; `if`(m=0, 1, `if`(i<1, 0,
             b(m, i-1)+`if`(l[i]>m, 0, b(m-l[i], i))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100); # Alois P. Heinz, Feb 05 2014

Mathematica

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 *)

PARI

isokp(p, n) = {for (k=1, #p, if ((p[k]==1) || (n % p[k]), return (0));); return (1);}
a(n) = {my(nb = 0); forpart(p=n, if (isokp(p,n), nb++)); nb;} \\ Michel Marcus, Jun 30 2015