A317385 -id:A317385 - cp's OEIS frontend

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

Alois P. Heinz, Jul 24 2018

nonn

			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.

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

Cf. A317240, A317242, A317385.

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);

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 *)

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

a(n) <= A317240(n).

A317390 A(n,k) is the n-th positive integer that has exactly k representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes; square array A(n,k), n>=1, k>=0, read by antidiagonals.

Original entry on oeis.org

2, 1, 5, 25, 3, 7, 43, 29, 4, 11, 211, 61, 37, 6, 15, 638, 261, 91, 40, 8, 23, 664, 848, 421, 111, 41, 9, 26, 1613, 1956, 921, 426, 121, 49, 10, 27, 2991, 3321, 2058, 969, 441, 124, 51, 12, 28, 7021, 3004, 3336, 2092, 1002, 484, 171, 52, 13, 31, 11306, 7162, 3319, 3368, 2094, 1026, 535, 184, 67, 14, 33

Offset: 1

Alois P. Heinz, Jul 27 2018

			A(6,2) = 49: 1 + 3 * (1 + 5 * (1 + 2)) = 1 + 2 * (1 + 23) = 49.
Square array A(n,k) begins:
   2,  1, 25,  43, 211,  638,  664, 1613, 2991, ...
   5,  3, 29,  61, 261,  848, 1956, 3321, 3004, ...
   7,  4, 37,  91, 421,  921, 2058, 3336, 3319, ...
  11,  6, 40, 111, 426,  969, 2092, 3368, 3554, ...
  15,  8, 41, 121, 441, 1002, 2094, 3741, 3928, ...
  23,  9, 49, 124, 484, 1026, 2283, 3914, 4846, ...
  26, 10, 51, 171, 535, 1106, 2381, 3979, 5552, ...
  27, 12, 52, 184, 540, 1156, 2388, 4082, 5886, ...
  28, 13, 67, 187, 591, 1191, 2432, 4126, 6293, ...

Columns k=0-10 give: A317242, A317391, A317392, A317393, A317394, A317395, A317396, A317397, A317398, A317399, A317400.
Row n=1 gives A317385.
A(n,n) gives A317537.
Cf. A317241.

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() local h, p, q; p, q:= proc() [] end, 0;
      proc(n, k)
        while nops(p(k))

b[n_, s_List] := 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 = q + 1; h = b[q, {}]; p[h] = Append[p[h], q]]; p[k][[n]]];
Table[Table[A[n, d - n], {n, 1, d}], {d, 1, 11}] // Flatten (* Jean-François Alcover, Dec 06 2019, from Maple *)

A317241(A(n,k)) = k.

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.

Original entry on oeis.org

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

Alois P. Heinz, Jul 26 2018

nonn

			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.

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

Cf. A317240, A317385.

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);

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 *)

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

A317537 The n-th 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

1, 29, 91, 426, 1002, 2283, 3979, 5886, 10861, 17116, 20749, 35106, 44031, 60919, 67453, 108655, 142429, 197107, 232625, 303317, 352093, 432517, 542935, 642520, 839938, 988791, 1050505, 1208559, 1612876, 1753324, 2129203, 2391496, 2735890, 3141916, 3593278

Offset: 1

Alois P. Heinz, Jul 30 2018

nonn

			a(1) = 1: 1.
a(2) = 29: 1 + 2 * (1 + 13) = 1 + 7 * (1 + 3) = 29.
a(3) = 91: 1 + 2 * (1 + 11 * (1 + 3)) = 1 + 3 * (1 + 29) = 1 + 5 * (1 + 17) = 91.

A diagonal of A317390.

Cf. A317385.

a(n) = A317390(n,n).

cp's OEIS Frontend

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.

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A317390 A(n,k) is the n-th positive integer that has exactly k representations of the form 1 + p1 * (1 + p2* ... * (1 + p_j)...), where [p1, ..., p_j] is a (possibly empty) list of distinct primes; square array A(n,k), n>=1, k>=0, read by antidiagonals.

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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.

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Maple

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A317537 The n-th 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.

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