A317241 Number of representations of n of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 0, 0, 2, 1, 0, 1, 0, 1, 0, 0, 2, 1, 1, 2, 2, 1, 3, 1, 1, 1, 0, 1, 2, 0, 2, 2, 1, 1, 1, 0, 0, 1, 1, 1, 3, 1, 0, 1, 1, 0, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 1, 0, 0, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Author

Alois P. Heinz, Jul 24 2018

Keywords

• nonn

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..65536

Crossrefs

Cf. A317240, A317242, A317385.

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:= n-> b(n, {}):
seq(a(n), n=1..200);

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_] := b[n, {}];
Array[a, 200] (* Jean-François Alcover, May 26 2019, after Alois P. Heinz *)

Formula

a(n) = 0 <=> n in { A317242 }.

a(n) <= A317240(n).