A309915 Total number of divisors d of m (counted with multiplicity), such that the prime signature of d is a partition of n and m runs through the set of least numbers whose prime signature is a partition of 2n.

1, 3, 16, 79, 371, 1683, 7413, 31769, 133692, 553848, 2265776, 9181670, 36928673, 147650125, 587734595, 2331625130, 9226486717, 36443758767, 143763811785, 566624864014, 2232055573265, 8789903797692, 34610963678036, 136287108614677, 536724439657635

Offset: 0

Author

Alois P. Heinz, Aug 22 2019

Keywords

• nonn

Examples

			a(2) = 16: The partitions of 2*2 are (4), (31), (22), (211), (1111). Least numbers with these prime signatures are 16, 24, 36, 60, 210. Their divisors with prime signatures (2) or (11) are {4}, {4,6}, {4,6,9}, {4,6,10,15}, {6,10,14,15,21,35}. The total number is 1 + 2 + 3 + 4 + 6.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..380

Crossrefs

Cf. A000041, A079025, A087443.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0 or i=1, (x+1)^n,
      b(n, i-1) +factor((x^(i+1)-1)/(x-1))*b(n-i, min(n-i, i))))
    end:
a:= n-> coeff(b(2*n$2), x, n):
seq(a(n), n=0..25);

Mathematica

b[n_, i_] := b[n, i] = Expand[If[n == 0 || i == 1, (x + 1)^n, b[n, i - 1] + Factor[(x^(i + 1) - 1)/(x - 1)] b[n - i, Min[n - i, i]]]];
a[n_] := Coefficient[b[2n, 2n], x, n];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

Formula

a(n) = A079025(2n,n).