A395091 a(n) is the number of partitions of n into distinct parts such that the sum of tau(k) over the parts k equals tau(n).

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 5, 1, 2, 2, 4, 1, 5, 1, 4, 2, 3, 1, 14, 1, 4, 1, 8, 1, 20, 1, 7, 2, 4, 1, 37, 1, 2, 2, 33, 1, 33, 1, 10, 14, 4, 1, 94, 1, 10, 1, 12, 1, 50, 2, 56, 1, 4, 1, 323, 1, 3, 22, 32, 1, 70, 1, 15, 2, 85, 1, 546, 1, 5, 23, 14, 1, 88, 1

Offset

0,5

Comments

We define A000005(0) = 0.

Links

Felix Huber, Table of n, a(n) for n = 0..10000

Formula

a(n) = [x^n*y^A000005(n)] Product_{k>=1} (1 + x^k*y^A000005(k)), where A000005(0) = 0.

a(p) = 1 for primes p.

1 <= a(n) <= A000009(n).

Example

a(4) = 2: [4], [1, 3], since tau(4) = 3 and tau(1) + tau(3) = 1 + 2 = 3.
a(12) = 5: [12], [2, 10], [1, 4, 7], [2, 3, 7], [1, 2, 9], since tau(12) = 6 and each listed partition has tau-sum 6.

Maple

A395091 := proc(n)
    local h, g, f;
    h := proc(k) option remember; `if`(k <= 0, 0, NumberTheory:-tau(k)) end proc:
    g := proc(k) option remember; `if`(k <= 0, 0, g(k - 1) + h(k)) end proc:
    f := proc(i, j, k) option remember;
        `if`(i = 0 and j = 0, 1, `if`(i < 0 or j < 0 or k = 0 or j > g(k), 0,
        f(i, j, k - 1) + f(i - k, j - h(k), k - 1)))
    end proc:
    `if`(n = 0, 1, f(n, h(n), n))
end proc:
seq(A395091(n), n = 0 .. 79);

Mathematica

a[n_]:=Count[Total/@DivisorSigma[0, Select[IntegerPartitions[n], DuplicateFreeQ]], DivisorSigma[0, n]]; a[0]=1; Array[a, 70, 0] (* James C. McMahon, May 01 2026 *)

Crossrefs

Cf. A000005, A000009, A394248, A395082, A395096, A395097, A395111.

Sequence in context: A330692 A349658 A345992 * A350333 A138010 A206487

Adjacent sequences: A395088 A395089 A395090 * A395092 A395093 A395094

Keyword

nonn

Author

Felix Huber, Apr 27 2026

Status

approved