A317384 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 (not necessarily distinct) primes.

2, 1, 13, 31, 43, 91, 111, 231, 175, 274, 351, 471, 703, 526, 463, 931, 823, 1723, 1579, 1279, 1903, 2083, 1791, 2143, 2227, 2443, 2671, 2781, 2335, 3807, 3163, 3631, 3199, 4243, 5314, 5482, 5107, 4671, 6231, 6681, 8863, 7483, 6111, 6331, 7879, 8031, 8023, 9351

Offset: 0

Author

Alois P. Heinz, Jul 26 2018

Keywords

• nonn

Examples

			a(1) = 1: 1.
a(2) = 13: 1 + 2 * (1 + 5) = 1 + 3 * (1 + 3) = 13.
a(3) = 31: 1 + 2 * (1 + 2 * (1 + 2 * (1 + 2))) = 1 + 3 * (1 + 3 * (1 + 2)) = 1 + 5 * (1 + 5) = 31.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A317240, A317385.

Programs

Maple

b:= proc(n) option remember; `if`(n=1, 1,
      add(b((n-1)/p), p=numtheory[factorset](n-1)))
    end:
a:= proc(n) option remember; local k;
      for k while n<>b(k) do od; k
    end:
seq(a(n), n=0..50);

Mathematica

pp[n_] := pp[n] = FactorInteger[n][[All, 1]];
q[n_] := q[n] = Switch[n, 1, True, 2, False, _, AnyTrue[pp[n-1], q[(n-1)/#]&]];
b[n_] := b[n] = Which[n == 1, 1, ! q[n], 0, True, Sum[b[(n-1)/p], {p, pp[n-1]}]];
a[n_] := Module[{k}, For[k = 1, True, k++, If[n == b[k], Return[k]]]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Dec 07 2023, after Alois P. Heinz *)

Formula

a(n) = min { j > 0 : A317240(j) = n }.