A317240 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 (not necessarily distinct) primes.

1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 0, 1, 2, 0, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 4, 1, 3, 2, 0, 1, 2, 1, 3, 2, 1, 1, 3, 0, 2, 3, 2, 1, 3, 1, 3, 3, 1, 2, 4, 1, 2, 1, 3, 1, 2, 1, 2, 3, 2, 1, 3, 1, 4, 2, 2, 1, 3, 1, 4, 3, 2, 1, 5, 3, 3, 4, 0, 2, 2, 1, 3, 2, 2, 1, 5, 1, 3

Offset: 1

Author

Alois P. Heinz, Jul 24 2018

Keywords

• nonn

Examples

			a(13) = 2: 1 + 2 * (1 + 5) = 1 + 3 * (1 + 3) = 13.
a(31) = 3: 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 = 1..65536

Crossrefs

Cf. A180337, A317241, A317384.

Programs

Maple

a:= proc(n) option remember; `if`(n=1, 1,
      add(a((n-1)/p), p=numtheory[factorset](n-1)))
    end:
seq(a(n), n=1..200);

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)/#]&]];
a[n_] := a[n] = Which[n == 1, 1, !q[n], 0, True, Sum[a[(n-1)/p], {p, pp[n-1]}]];
Array[a, 105] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{prime p|(n-1)} a((n-1)/p) for n>1, a(1) = 1.

a(n) = 0 <=> n in { A180337 }.

a(n) >= A317241(n).

G.f. A(x) satisfies: A(x) = x * (1 + A(x^2) + A(x^3) + A(x^5) + ... + A(x^prime(k)) + ...). - Ilya Gutkovskiy, May 09 2019