A395111 - OEIS

%I #13 Apr 26 2026 23:54:40

%S 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,

%T 64,72,1,88,98,109,4,134,147,163,10,195,216,236,53,35,310,339,1,3,5,

%U 480,188,572,1,675,32,796,865,937,352,1100,1189,131,1,1502,1622

%N 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).

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

%C 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.

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

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

%C We define A007947(0) = 0.

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

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

%H Felix Huber, <a href="/A395111/b395111.txt">Table of n, a(n) for n = 0..1000</a>

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a395/A395111.java">Java program</a> (github)

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

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

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

%F a(A000079(n)) = 1.

%e a(8) = 1: [8], since rad(8) = 2 and this is the only distinct partition with radical sum 2.

%e 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.

%e 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.

%p A := proc(f, N)

%p     local d, k, n, s, v;

%p     v := Vector(N);

%p     for n from 1 to N do

%p         v[n] := f(n);

%p     end do;

%p     d := Array(0 .. N, 0 .. N, 'fill = 0');

%p     d[0, 0] := 1;

%p     for k from 1 to N do

%p         for n from N by -1 to k do

%p             for s from N by -1 to v[k] do

%p                 d[n, s] := d[n, s] + d[n - k, s - v[k]];

%p             end do;

%p         end do;

%p     end do;

%p     n -> `if`(n = 0, 1, d[n, v[n]]);

%p end proc:

%p N := 66:

%p A395111 := A(NumberTheory:-Radical, N):

%p seq(A395111(n), n = 0 .. N);

%t 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 *)

%o (PARI) rad(n) = factorback(factorint(n)[, 1]);

%o 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

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

%K nonn

%O 0,4

%A _Felix Huber_, Apr 15 2026