A381897 a(n) = least integer m >= 2 such that prime(n) is a sum of the form Sum_{k>=0} floor(h/m^k) for some integer h >= 1.

3, 2, 3, 2, 2, 3, 3, 2, 2, 4, 2, 4, 2, 4, 2, 2, 3, 3, 2, 2, 2, 2, 5, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 2, 3, 2, 2, 2, 2, 4, 3, 3, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 2, 2, 3, 2, 4, 2, 3, 3, 3, 2, 4, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 4, 2, 4

Offset: 1

Author

Clark Kimberling, Mar 09 2025

Keywords

• nonn

Examples

			a(10) = 4 because 4 is the least m such that prime(10) is a sum of the form Sum_{k>=0} [h/m^k] for some h >= 1; that sum is 29 = [23/1] + [23/4] + [23/16], where [ ] = floor.
		

Crossrefs

Cf. A000040, A381239, A382278.

Programs

Mathematica

f[h_, m_] := Sum[Floor[h/m^k], {k, 0, Floor[Log[m, h]]}]
{rng, n} = {1000, 6};
Table[u[m] = Select[Range[rng], PrimeQ[f[#, m]] &], {m, 2, n}];
tmp = SortBy[Map[#[[1]] &, GatherBy[Flatten[Table[
       Transpose[{ConstantArray[m, Length[u[m]]],
         Map[PrimePi[f[#, m]] &, u[m]]}], {m, 2, n}],1], #[[2]] &]], #[[2]] &];
tmp = Map[#[[1]] &, Take[tmp, Position[Differences[Map[#[[2]] &, tmp]], x_ /; x != 1, 1, 1][[1]][[1]]]]
(* Peter J. C. Moses, Feb 19 2025 *)

Formula

a(n) = A382278(prime(n)). - Pontus von Brömssen, Mar 22 2025