A093802 Number of distinct factorizations of 105*2^n.

5, 15, 36, 74, 141, 250, 426, 696, 1106, 1711, 2593, 3852, 5635, 8118, 11548, 16231, 22577, 31092, 42447, 57464, 77213, 103009, 136529, 179830, 235514, 306751, 397506, 512607, 658030, 841020, 1070490, 1357195, 1714274, 2157539, 2706174, 3383187, 4216358

Offset: 0

Author

Alford Arnold, May 19 2004

Keywords

• easy
• nonn

Examples

			105*A000079 is 105, 210, 420, 840, 1680, 3360, ... and there are 15 distinct factorizations of 210 so a(1) = 15.
a(0) = 5: 105*2^0 = 105 = 3*5*7 = 3*35 = 5*21 = 7*15. - _Alois P. Heinz_, May 26 2013
		

Links

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

Crossrefs

Similar sequences: 45*A000079 => A002763, [1, 3, 9, 27, 81, 243...]*A000079 => A054225, 1*A002110 => A000110, 2*A002110 => A035098, A000142 => A076716.

Cf. A001055, A126442, A129306, A346823.

Column k=3 of A346426.

Programs

Maple

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= n-> b((105*2^n)$2):
seq(a(n), n=0..50);  # Alois P. Heinz, May 26 2013

Mathematica

b[n_, k_] := b[n, k] = If[n > k, 0, 1] + If[PrimeQ[n], 0,
     Sum[If[d > k, 0, b[n/d, d]], {d, Divisors[n][[2;;-2]]}]];
a[n_] := b[105*2^n, 105*2^n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 15 2021, after Alois P. Heinz *)

Extensions

2 more terms from Alford Arnold, Aug 29 2007

Corrected offset and extended beyond a(7) by Alois P. Heinz, May 26 2013