A394248 a(n) is the number of partitions of n into distinct parts such that the sum of sigma(k) over the parts k equals sigma(n).

1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 6, 6, 1, 1, 8, 9, 13, 3, 1, 2, 1, 7, 17, 23, 22, 1, 1, 36, 33, 5, 1, 2, 1, 20, 56, 82, 1, 1, 25, 52, 112, 60, 1, 14, 106, 25, 177, 259, 1, 1, 1, 378, 372, 117, 234, 27, 1, 288, 505, 130, 1, 1, 1, 1043

Offset

0,4

Comments

For a prime p we have sigma(p) = p + 1. Suppose p = x_1 + ... + x_k is a partition of p into distinct parts with k >= 2. If none of the parts equals 1, then sigma(x_i) >= x_i + 1 for all i, hence Sum_{i=1..k} sigma(x_i) >= p + k >= p + 2 > sigma(p). If 1 is a part, write p = 1 + y_1 + ... + y_{k-1}. Then Sum_{i=1..k} sigma(x_i) >= p + (k - 1), and equality with sigma(p) = p + 1 can occur only if k = 2 and y_1 is prime, which forces p = 3. Hence a(p) = 1 for all primes p other than 3, and a(3) = 2.

We define A000203(0) = 0.

Links

Felix Huber, Table of n, a(n) for n = 0..500

Formula

a(n) = [x^n*y^A000203(n)] Product_{k >= 1} (1 + x^k*y^A000203(k)), where A000203(0) = 0.

a(p) = 1 for primes p other than 3.

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

Example

a(3) = 2: [3], [1, 2], since sigma(3) = 4 = sigma(1) + sigma(2).
a(26) = 9: [26], [5, 6, 15], [3, 4, 9, 10], [1, 7, 8, 10], [3, 6, 7, 10], [1, 2, 4, 9, 10], [1, 2, 6, 7, 10], [2, 3, 4, 8, 9], [2, 3, 6, 7, 8], since sigma(26) = 42 and each listed partition has sum of sigma values 42.

Maple

A394248 := proc(n)
    local h, g, f;
    h := proc(k) option remember; `if`(k <= 0, 0, NumberTheory:-sigma(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(A394248(n), n = 0 .. 74);

Mathematica

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

Crossrefs

Cf. A000009, A000203, A395082, A395091, A395096, A395097, A395111.

Sequence in context: A345971 A211355 A211353 * A094189 A122771 A217710

Adjacent sequences: A394244 A394245 A394247 * A394250 A394251 A394252

Keyword

nonn

Author

Felix Huber, Apr 27 2026

Status

approved