A086146 a(n) is the smallest k>=n such that the number of partitions of k is a multiple of n, or -1 if no such k exists.

1, 2, 3, 11, 7, 9, 10, 11, 14, 19, 12, 21, 28, 19, 24, 66, 54, 21, 20, 58, 24, 25, 32, 70, 44, 28, 39, 55, 91, 97, 44, 66, 35, 94, 39, 80, 86, 47, 129, 66, 45, 75, 100, 58, 129, 75, 56, 70, 68, 74, 178, 62, 66, 340, 58, 75, 209, 97, 93, 124, 115, 101, 138, 66, 84, 75, 111, 94

Offset: 1

Author

Jim Nastos, Aug 25 2003

Keywords

• nonn

Comments

I do not know if a(n) exists for all n. First term which is currently unknown is a(2219) (a(2219) > 11600, while a(2218) = 2602).

Examples

			a(4) is 11 because 11 is the smallest number for which P(11) is divisible by 4, where P() is the partition function.
		

Crossrefs

Cf. A046641.

Programs

Maple

for i from 2 while i < 30000 do for j from i while j < 1000000000 do c := numbpart(j); if (c mod i = 0) then print(i,j); break; end if; end do; end do;
# alternative
A086146 := proc(n)
    local k ;
    for k from n do
        if combinat[numbpart](k) mod n =0 then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Apr 22 2013

Mathematica

kmax = 10^9;
a[n_] := Module[{k}, For[k = n, k <= kmax, k++, If[Divisible[ PartitionsP[k], n], Return[k]]]] /. Null -> -1;
Table[a[n], {n, 1, 68}] (* Jean-François Alcover, Jul 08 2024 *)

Extensions

More terms from Ray Chandler, Sep 16 2003