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

1,1

Links

Alois P. Heinz, Antidiagonals n = 1..34, flattened

Index entries for sequences that are permutations of the natural numbers

Formula

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

Example

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, ...

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))<n do q:= q+1;
          h:= b(q, {});
          p(h):= [p(h)[], q]
        od; p(k)[n]
      end
    end():
seq(seq(A(n, d-n), n=1..d), d=1..10);

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 *)

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.

Sequence in context: A349852 A260701 A184300 * A370163 A075403 A260503

Adjacent sequences: A317387 A317388 A317389 * A317391 A317392 A317393

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 27 2018

Status

approved