A385374 a(n) is the number of partitions of n into tau(n) distinct parts.

1, 0, 1, 0, 2, 0, 3, 0, 3, 1, 5, 0, 6, 5, 6, 1, 8, 0, 9, 0, 27, 34, 11, 0, 40, 64, 72, 14, 14, 0, 15, 44, 150, 169, 185, 0, 18, 249, 270, 5, 20, 11, 21, 454, 532, 478, 23, 0, 176, 1057, 672, 1360, 26, 288, 864, 434, 972, 1033, 29, 0, 30, 1285, 4494, 4011, 1495

Offset: 1

Author

Felix Huber, Jul 06 2025

Keywords

• nonn
• look

Examples

			a(14) = 5 because there are 5 partitions of 14 into tau(14) = 4 distinct parts: [1, 2, 3, 8], [1, 2, 4, 7], [1, 2, 5, 6], [1, 3, 4, 6], [2, 3, 4, 5].
		

Links

• Felix Huber, Table of n, a(n) for n = 1..4000

Crossrefs

Subsequence of A060016.

Cf. A000005, A385375.

Programs

Maple

b:= proc(n,i,k)
    option remember;
    if n=0 and k=0 then
        return 1
    elif n=0 or k=0 or i<1 then
        return 0
    elif i<=n then
        return b(n,i-1,k)+b(n-i,i-1,k-1)
    else
        return b(n,i-1,k)
    fi;
end proc:
A385374:=n->b(n,n,NumberTheory:-tau(n));
seq(A385374(n),n=1..65);

Mathematica

a[n_]:=Length[Select[Union/@IntegerPartitions[n,{DivisorSigma[0,n]}],Length[#]==DivisorSigma[0,n]&]];Array[a,65] (* James C. McMahon, Jul 11 2025 *)