A254578 Number of ordered factorizations into distinct factors.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 13, 1, 3, 3, 5, 1, 13, 1, 5, 3, 3, 3, 13, 1, 3, 3, 13, 1, 13, 1, 5, 5, 3, 1, 21, 1, 5, 3, 5, 1, 13, 3, 13, 3, 3, 1, 29, 1, 3, 5, 11, 3, 13, 1, 5, 3, 13, 1, 29, 1, 3, 5, 5, 3, 13, 1, 21, 3, 3

Offset: 1

Author

Geoffrey Critzer, Feb 01 2015

Keywords

• nonn

Examples

			a(20)=5 because there are 5 ordered factorizations of 20 into distinct factors: 2*10, 4*5, 5*4, 10*2, 20.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).

Crossrefs

Cf. A074206, A254577, A251683.

Programs

Maple

with(numtheory):
b:= proc(n, i, p) option remember; `if`(n<=i, (p+1)!, 0)+add(
      b(n/d, d-1, p+1), d=select(x->x<=i, divisors(n)minus{1, n}))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 01 2015

Mathematica

f[n_] := f[n] = Level[Table[Map[Prepend[#, d] &, f[n/d]], {d,Rest[Divisors[n]]}], {2}];
f[1] = {{}};
Map[Length,Map[Select[#, Apply[Unequal, #] &] &, Table[f[n], {n, 1, 60}]]]