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.

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

Author

Alois P. Heinz, Jul 27 2018

Keywords

• nonn
• tabl

Examples

			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, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 1..34, flattened
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

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.

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

Mathematica

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 *)

Formula

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