A266325 Smallest integer m such that there is a partition of m with product of multiplicities of parts equal to n.

0, 2, 3, 4, 5, 6, 7, 8, 9, 9, 11, 10, 13, 11, 11, 12, 17, 12, 19, 13, 13, 15, 23, 14, 15, 17, 15, 15, 29, 16, 31, 16, 17, 21, 17, 17, 37, 23, 19, 18, 41, 19, 43, 19, 19, 27, 47, 20, 21, 20, 23, 21, 53, 21, 21, 21, 25, 33, 59, 22, 61, 35, 22, 22, 23, 23, 67, 25

Offset: 1

Author

Emeric Deutsch and Alois P. Heinz, Jan 04 2016

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2500

Crossrefs

Cf. A266477.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, `if`(p=1, 1, 0),
      `if`(i<1, 0, b(n, i-1, p)+add(`if`(irem(p, j)=0,
       b(n-i*j, i-1, p/j), 0), j=1..n/i)))
    end:
a:= proc(n) option remember; local m;
      if isprime(n) then return n fi;
      for m from 0 do if b(m$2, n)>0 then return m fi od
    end:
seq(a(n), n=1..100);

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, If[p == 1, 1, 0], If[i < 1, 0, b[n, i - 1, p] + Sum[If[Mod[p, j] == 0, b[n - i*j, i - 1, p/j], 0], {j, 1, n/i}]]]; a[n_] := a[n] = Module[{m}, If[PrimeQ[n], Return[n]]; For[m = 0, True, m++, If[b[m, m, n] > 0, Return[m]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 21 2016, translated from Maple *)

Formula

a(n) = min { m >= 0 : A266477(m,n) > 0 }.

p in primes => a(p) = p.