A395096 a(n) is the number of partitions of n into distinct parts such that the sum of A001221 over the parts equals A001221(n).

1, 1, 1, 2, 2, 2, 3, 1, 2, 2, 5, 1, 7, 1, 6, 7, 1, 2, 6, 1, 8, 10, 9, 1, 7, 1, 7, 1, 9, 1, 38, 1, 2, 10, 12, 12, 13, 1, 7, 10, 12, 1, 60, 1, 9, 12, 12, 1, 10, 1, 10, 12, 12, 1, 10, 12, 12, 13, 11, 1, 89, 1, 9, 11, 1, 9, 92, 1, 10, 14, 94, 1, 12, 1, 10, 14, 15, 12

Offset

0,4

Comments

We define A001221(0) = 0.

Links

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

Formula

a(n) = [x^n*y^A001221(n)] Product_{k>=1} (1 + x^k*y^A001221(k)), where A001221(0) = 0.

For m = p^k a prime power, a(m) = 2 if A001221(m-1) = 1, and 1 otherwise.

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

Example

a(6) = 3: [6], [2, 4], [1, 2, 3], since A001221(6) = 2 and these are exactly the three distinct partitions with A001221 sum 2.

Maple

A := proc(f, N)
    local d, i, k, m, n, s, v;
    v := [0, seq(f(i), i = 1 .. N)];
    m := max(op(v));
    d := Array(0 .. N, 0 .. m, 'fill = 0');
    d[0, 0] := 1;
    for k from 1 to N do
        for n from N by -1 to k do
            for s from m by -1 to v[k + 1] do
                d[n, s] := d[n, s] + d[n - k, s - v[k + 1]]
            od
        od
    od;
    [seq(d[n, v[n + 1]], n = 0 .. N)]
end proc:
N := 77: # Enlarge if you want more terms
w := n -> NumberTheory:-Omega(n, 'distinct'):
A395096 := A(w, N):
A395096(N);

Mathematica

{1}~Join~Table[With[{r = PrimeNu[n]}, Count[Select[IntegerPartitions[n], DuplicateFreeQ], _?(Total[PrimeNu /@ #] == r &)]], {n, 60}] (* Michael De Vlieger, Apr 15 2026 *)

Crossrefs

Cf. A000009, A001221, A395097, A395111.

Sequence in context: A256915 A348367 A209254 * A376307 A227738 A103960

Adjacent sequences: A395093 A395094 A395095 * A395097 A395098 A395099

Keyword

nonn

Author

Felix Huber, Apr 15 2026

Status

approved