A211111 Number of partitions of n into distinct divisors > 1 of n.

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 19, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 14, 1

Offset: 0

Author

Reinhard Zumkeller, Apr 01 2012

Keywords

• nonn

Comments

a(A136446(n)) > 1.

Examples

			n=12: the divisors > 1 of 12 are {2,3,4,6,12}, there are exactly two subsets which sum up to 12, namely {12} and {2,4,6}, therefore a(12) = 2;
a(13) = #{13} = 1, because 13 is prime, having no other divisor > 1;
n=14: the divisors > 1 of 14 are {2,7,14}, {14} is the only subset summing up to 14, therefore a(14) = 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n=1..1000 from Reinhard Zumkeller)

Crossrefs

Cf. A211110, A033630, A027750.

Cf. A065205, A136446.

Programs

Haskell

a211111 n = p (tail $ a027750_row n) n where
   p _  0 = 1
   p [] _ = 0
   p (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-1))))
          end; forget(b):
      b(n, nops(l))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 18 2021

Mathematica

a[n_] := Count[IntegerPartitions[n, All, Divisors[n] // Rest], P_ /; Reverse[P] == Union[P]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 18 2021 *)

Extensions

a(0)=1 prepended by Alois P. Heinz, Nov 18 2021