A317385 Smallest positive integer that has exactly n representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

2, 1, 25, 43, 211, 638, 664, 1613, 2991, 7021, 11306, 9439, 17361, 23230, 40886, 38341, 49063, 36583, 99111, 111229, 110631, 171718, 233451, 255531, 309141, 327643, 369519, 521266, 489406, 738544, 682690, 812826, 1048594, 1015096, 2003002, 2118439, 1602360, 2204907, 2850772, 2702743, 2794198

Offset: 0

Author

Alois P. Heinz, Jul 26 2018

Keywords

• nonn

Examples

			a(1) = 1: 1.
a(2) = 25: 1 + 2 * (1 + 11) = 1 + 3 * (1 + 7) = 25.
a(3) = 43: 1 + 2 * (1 + 5 * (1 + 3)) = 1 + 3 * (1 + 13) = 1 + 7 * (1 + 5) = 43.
		

Crossrefs

Row n=1 of A317390.

Cf. A317241, A317384.

Programs

Maple

b:= proc(n, s) option remember; `if`(n=1, 1, add(b((n-1)/p
      , s union {p}), p=numtheory[factorset](n-1) minus s))
    end:
a:= proc(n) option remember; local k;
      for k while n<>b(k, {}) do od; k
    end:
seq(a(n), n=0..15);

Mathematica

b[n_, s_] := b[n, s] = If[n == 1, 1, Sum[If[p == 1, 0, b[(n - 1)/p, s~Union~{p}]], {p, FactorInteger[n - 1][[All, 1]]~Complement~s}]];
A[n_, k_] := Module[{h, p, q = 0}, p[_] = {}; While[Length[p[k]] < n, q++; h = b[q, {}]; p[h] = Append[p[h], q]]; p[k][[n]]];
a[k_] := If[k == 0, 2, A[1, k]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz in A317390 *)

Formula

a(n) = min { j > 0 : A317241(j) = n }.