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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A184298 A260701 A184300 * A075403 A260503 A236436

Adjacent sequences:  A317387 A317388 A317389 * A317391 A317392 A317393

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 27 2018

STATUS

approved

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Last modified May 9 20:59 EDT 2021. Contains 343746 sequences. (Running on oeis4.)