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

1,5

Links

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

Example

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].

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 *)

Crossrefs

Subsequence of A060016.

Cf. A000005, A385375.

Sequence in context: A035376 A259708 A029220 * A249901 A253274 A365920

Adjacent sequences: A385371 A385372 A385373 * A385375 A385376 A385377

Keyword

nonn,look

Author

Felix Huber, Jul 06 2025

Status

approved