A393359 a(n) is the number of ways to successively choose subsets of sizes equal to the proper divisors of n from [n], each choice taken from the remaining elements.

1, 2, 3, 12, 5, 60, 7, 840, 504, 7560, 11, 0, 13, 360360, 2522520, 10810800, 17, 0, 19, 0, 465585120, 349188840, 23, 0, 1062600, 8923714800, 57366738000, 14456417976000, 29, 0, 31, 6498159880212000, 5662884427200, 4760556688800, 15200373988800, 0, 37, 102748681866600, 485721041551200, 0

Offset

1,2

Comments

If the sum of the proper divisors of n exceeds n (i.e., n is abundant), the process cannot be completed and a(n) = 0.

Equivalently, for sigma(n) <= 2n, a(n) is the number of ordered set partitions of [n] into parts of sizes equal to the proper divisors of n together with one additional part of size n minus their sum.

For perfect n, the remainder part has size 0 and a(n) = n!/Product_{d|n, d<n} d!. Thus perfect numbers are exactly those n for which the selection process exhausts [n] without remainder.

Links

Felix Huber, Table of n, a(n) for n = 1..1364

Formula

a(n) = 0 if sigma(n) > 2*n, otherwise:

a(n) = Product_{i=1..tau(n)-1} binomial(n - Sum_{j=1..i-1} d_j, d_i), where d_1 > ... > d_{tau(n)-1} are the proper divisors of n.

a(n) = n!/((2*n - sigma(n))!*Product_{d|n, d<n} d!).

a(p) = p for primes p.

Example

a(6) = binomial(6, 3)*binomial(3, 2)*binomial(1, 1) = 60 (perfect).
a(10) = binomial(10, 5)*binomial(5, 2)*binomial(3, 1) = 7560 (deficient).
a(11) = binomial(11, 1) = 11 (prime).
a(12) = 0 since sigma(12) = 28 > 24 (abundant).

Maple

A393359 := proc(n)
    local d, k, m, a;
    d := NumberTheory:-Divisors(n);
    m := n;
    a := 1;
    for k from nops(d) - 1 by -1 to 1 do
        m := m - d[k];
        if m < 0 then return 0 end if;
        a := a*binomial(m + d[k], d[k])
    end do;
    a
end proc:
seq(A393359(n), n = 1 .. 40);

Crossrefs

Cf. A000005, A000203, A007318, A007837, A023645, A027750, A038041.

Sequence in context: A288058 A281850 A282216 * A245678 A124444 A038610

Adjacent sequences: A393356 A393357 A393358 * A393360 A393361 A393362

Keyword

nonn

Author

Felix Huber, Mar 28 2026

Status

approved