A382278 -id:A382278 - cp's OEIS frontend

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

Clark Kimberling, Mar 09 2025

nonn

			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.

Cf. A000040, A381239, A382278.

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 *)

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

A382324 a(n) = least integer h >= 1 such that n is a sum of the form Sum_{k>=0} floor(h/m^k) for some integer m >= 2.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 4, 5, 7, 6, 7, 10, 9, 10, 8, 9, 12, 10, 11, 17, 15, 12, 13, 19, 14, 15, 19, 20, 23, 21, 16, 17, 26, 18, 19, 26, 29, 20, 21, 27, 22, 23, 33, 30, 31, 24, 25, 33, 26, 27, 42, 40, 28, 29, 38, 30, 31, 40, 41, 47, 42, 43, 32, 33, 50, 34, 35, 47

Offset: 1

Clark Kimberling, Apr 01 2025

nonn

			a(12) = 10, because 10 is the least h such that 12 is a sum of the form Sum_{k>=0} floor(h/m^k) for some m >= 2; that sum is [10/1] + [10/4], where [ ] = floor.

Cf. A382278.

findH[n_, m_] := With[{hMin = Floor[(n*(m - 1) + m)/m], hMax = 2*n*m},
   SelectFirst[Range[hMin, hMax], Total[IntegerDigits[#1, m]] == m*#1 - n*(m - 1) &, None]];
aPair[n_] := With[{m = SelectFirst[Range[2, n + 1], findH[n, #1] =!= None &, None]},
If[m === None, None, {m, findH[n, m]}]];
t = ({#1, aPair[#1]} &) /@ Range[100]
Map[Last, Map[Last, t]]
(* Peter J. C. Moses, Mar 20 2025 *)

cp's OEIS Frontend

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.

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