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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.

%I A086146 #14 Jul 08 2024 09:00:07
%S A086146 1,2,3,11,7,9,10,11,14,19,12,21,28,19,24,66,54,21,20,58,24,25,32,70,
%T A086146 44,28,39,55,91,97,44,66,35,94,39,80,86,47,129,66,45,75,100,58,129,75,
%U A086146 56,70,68,74,178,62,66,340,58,75,209,97,93,124,115,101,138,66,84,75,111,94
%N 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.
%C A086146 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).
%e A086146 a(4) is 11 because 11 is the smallest number for which P(11) is divisible by 4, where P() is the partition function.
%p A086146 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;
%p A086146 # alternative
%p A086146 A086146 := proc(n)
%p A086146     local k ;
%p A086146     for k from n do
%p A086146         if combinat[numbpart](k) mod n =0 then
%p A086146             return k;
%p A086146         end if;
%p A086146     end do:
%p A086146 end proc: # _R. J. Mathar_, Apr 22 2013
%t A086146 kmax = 10^9;
%t A086146 a[n_] := Module[{k}, For[k = n, k <= kmax, k++, If[Divisible[ PartitionsP[k], n], Return[k]]]] /. Null -> -1;
%t A086146 Table[a[n], {n, 1, 68}] (* _Jean-François Alcover_, Jul 08 2024 *)
%Y A086146 Cf. A046641.
%K A086146 nonn
%O A086146 1,2
%A A086146 _Jim Nastos_, Aug 25 2003
%E A086146 More terms from _Ray Chandler_, Sep 16 2003