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

0,1

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A317240, A317385.

Sequence in context: A174170 A264373 A217490 * A247601 A013020 A012906

Adjacent sequences: A317381 A317382 A317383 * A317385 A317386 A317387

Keyword

nonn

Author

Alois P. Heinz, Jul 26 2018

Status

approved