A317384 -id:A317384 - 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

A317385 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 distinct primes.

Original entry on oeis.org

2, 1, 25, 43, 211, 638, 664, 1613, 2991, 7021, 11306, 9439, 17361, 23230, 40886, 38341, 49063, 36583, 99111, 111229, 110631, 171718, 233451, 255531, 309141, 327643, 369519, 521266, 489406, 738544, 682690, 812826, 1048594, 1015096, 2003002, 2118439, 1602360, 2204907, 2850772, 2702743, 2794198

Offset: 0

Alois P. Heinz, Jul 26 2018

nonn

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

Row n=1 of A317390.

Cf. A317241, A317384.

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:= proc(n) option remember; local k;
      for k while n<>b(k, {}) do od; k
    end:
seq(a(n), n=0..15);

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_, k_] := Module[{h, p, q = 0}, p[_] = {}; While[Length[p[k]] < n, q++; h = b[q, {}]; p[h] = Append[p[h], q]]; p[k][[n]]];
a[k_] := If[k == 0, 2, A[1, k]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz in A317390 *)

a(n) = min { j > 0 : A317241(j) = n }.

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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