A395111 a(n) is the number of partitions of n into distinct parts such that the sum of rad(k) over the parts k equals rad(n).

1, 1, 1, 2, 1, 2, 3, 3, 1, 2, 5, 6, 4, 8, 9, 10, 1, 14, 2, 18, 1, 24, 27, 30, 1, 3, 42, 1, 16, 59, 64, 72, 1, 88, 98, 109, 4, 134, 147, 163, 10, 195, 216, 236, 53, 35, 310, 339, 1, 3, 5, 480, 188, 572, 1, 675, 32, 796, 865, 937, 352, 1100, 1189, 131, 1, 1502, 1622

Offset

0,4

Comments

Combines the additive structure of partitions into distinct parts with the multiplicative radical function A007947.

For squarefree n (A005117), reduces to partitions into distinct squarefree parts (A087188), since rad(n) = n and rad(k) <= k forces rad(k) = k for all parts.

Every admissible part k has all its prime divisors dividing n, since otherwise rad(k) would contain a prime not dividing rad(n).

If n is a power of 2, the radical sum condition forces the unique partition [n].

We define A007947(0) = 0.

Conjecture 1: The set of values {a(n)} coincides with the set of values {A087188(n)}.

Conjecture 2: The set {n : a(n) = 1} has natural density 0.

Links

Felix Huber, Table of n, a(n) for n = 0..1000

Sean A. Irvine, Java program (github)

Formula

a(n) = [x^n*y^A007947(n)] Product_{k>=1} (1 + x^k*y^A007947(k)), where A007947(0) = 0.

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

a(A005117(n)) = A087188(A005117(n)) for n >= 1.

a(A000079(n)) = 1.

Example

a(8) = 1: [8], since rad(8) = 2 and this is the only distinct partition with radical sum 2.
a(10) = 5: [10], [3, 7], [1, 2, 7], [1, 3, 6], [2, 3, 5], since rad(10) = 10 and these are exactly the five distinct partitions with radical sum 10.
a(12) = 4: [12], [3, 9], [1, 3, 8], [1, 2, 9], since rad(12) = 6 and these are exactly the four distinct partitions with radical sum 6.

Maple

A := proc(f, N)
    local d, k, n, s, v;
    v := Vector(N);
    for n from 1 to N do
        v[n] := f(n);
    end do;
    d := Array(0 .. N, 0 .. N, '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 N by -1 to v[k] do
                d[n, s] := d[n, s] + d[n - k, s - v[k]];
            end do;
        end do;
    end do;
    n -> `if`(n = 0, 1, d[n, v[n]]);
end proc:
N := 66:
A395111 := A(NumberTheory:-Radical, N):
seq(A395111(n), n = 0 .. N);

Mathematica

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Table[With[{r = rad[n]}, Count[Select[IntegerPartitions[n], DuplicateFreeQ], _?(Total[rad /@ #] == r &)]], {n, 0, 60}] (* Michael De Vlieger, Apr 15 2026 *)

Prog

(PARI) rad(n) = factorback(factorint(n)[, 1]);
a(n) = my(nb=0); forpart(p=n, if ((#Set(p) == #p) && sum(k=1, #p, rad(p[k])) == rad(n), nb++)); nb; \\ Michel Marcus, Apr 15 2026

Crossrefs

Cf. A000009, A000079, A005117, A007947, A087188, A395096, A395097.

Sequence in context: A328471 A227909 A301984 * A210805 A303842 A057041

Adjacent sequences: A395108 A395109 A395110 * A395112 A395113 A395114

Keyword

nonn

Author

Felix Huber, Apr 15 2026

Status

approved