A180337 -id:A180337 - cp's OEIS frontend

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

Alois P. Heinz, Jul 24 2018

nonn

			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.

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

Cf. A180337, A317241, A317384.

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);

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 *)

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

A317242 Positive integers having no representation of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes.

Original entry on oeis.org

2, 5, 7, 11, 15, 23, 26, 27, 28, 31, 33, 35, 36, 47, 50, 56, 57, 63, 66, 78, 81, 82, 95, 96, 106, 116, 119, 120, 122, 129, 136, 156, 162, 166, 167, 190, 193, 215, 218, 219, 227, 236, 244, 254, 263, 286, 289, 330, 335, 342, 352, 359, 387, 393, 395, 396, 414

Offset: 1

Alois P. Heinz, Jul 24 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Column k=0 of A317390.

Cf. A180337 (subsequence), A317241.

q:= proc(n, s) option remember; is (n=1 or ormap(p->
      q((n-1)/p, s union {p}), numtheory[factorset](n-1) minus s))
    end:
a:= proc(n) option remember; local k; for k from
     `if`(n=1, 2, 1+a(n-1)) while q(k, {}) do od; k
    end:
seq(a(n), n=1..100);

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}]];
Position[Array[b[#, {}]&, 10^5], 0] // Flatten (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz in A317241 *)

A317241(a(n)) = 0.

cp's OEIS Frontend

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.

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