A092976 Consider all partitions of n into parts all of which are divisors of n; a(n) = number of distinct values taken by the product of the parts.

1, 1, 2, 2, 3, 2, 7, 2, 5, 4, 10, 2, 19, 2, 13, 13, 9, 2, 37, 2, 29, 17, 19, 2, 61, 6, 22, 10, 39, 2, 247, 2, 17, 25, 28, 25, 127, 2, 31, 29, 97, 2, 450, 2, 59, 82, 37, 2, 217, 8, 146, 37, 69, 2, 271, 37, 133, 41, 46, 2, 1558, 2, 49, 112, 33, 43, 1038, 2, 89, 49, 1105, 2, 469, 2, 58, 211, 99, 49, 1423, 2, 353

Offset

0,3

Comments

a(n) > tau(n) + A(n) + R(n), where tau(n) = number of divisors of n, A(n) = product of powers of nontrivial divisors whose sum with multiplicity is < n and R(n) = numbers of the form r^(k) > n where r is a divisor of n and k <= n/r.

Links

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

Formula

a(n) = 2 <=> n in { A000040 }. - Alois P. Heinz, Oct 28 2021

Example

a(10) = 10, the numbers arising are 1,2,4,5,8,10,16,20,25 and 32; e.g. 25 = 5*5, 8 = 2*2*2*1*1*1*1, 32 = 2*2*2*2*2, etc.

Maple

with(numtheory):
a:= proc(n) local b, l, s;
      l:= sort([divisors(n)[]]);
      b:= proc(n, i, p)
            if n<0 then
          elif n=0 then s:= s union {p}
          elif i=0 then
          else b(n-l[i], i, p*l[i]); b(n, i-1, p)
            fi
          end;
      s:= {};
      b(n, nops(l), 1);
      nops(s)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 21, 2011

Mathematica

a[n_] := Module[{ b, l, s}, l = Divisors[n]; b[m_, i_, p_] := Which[m<0, , m == 0, s = Union[s, {p}], i == 0, , True, b[m - l[[i]], i, p*l[[i]]]; b[m, i-1, p]]; s = {}; b[n, Length[l], 1]; Length[s]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000040, A092975.

Sequence in context: A294933 A015996 A256564 * A084705 A051886 A244080

Adjacent sequences: A092973 A092974 A092975 * A092977 A092978 A092979

Keyword

nonn

Author

Amarnath Murthy, Mar 27 2004

Extensions

More terms from Vladeta Jovovic, Aug 06 2005

More terms from Alois P. Heinz, Feb 21 2011

a(0)=1 prepended by Alois P. Heinz, Oct 28 2021

Status

approved