A378193 Rectangular array read by descending antidiagonals: row n shows the integers m such that the number of Pythagorean primes (including multiplicities) that divide m is n-1.

1, 2, 5, 3, 10, 25, 4, 13, 50, 125, 6, 15, 65, 250, 625, 7, 17, 75, 325, 1250, 3125, 8, 20, 85, 375, 1625, 6250, 15625, 9, 26, 100, 425, 1875, 8125, 31250, 78125, 11, 29, 130, 500, 2125, 9375, 40625, 156250, 390625, 12, 30, 145, 650, 2500, 10625, 46875, 203125, 781250, 1953125

Offset: 1

Author

Clark Kimberling, Jan 14 2025

Keywords

• nonn
• tabl

Comments

Every positive integer occurs exactly once.

Examples

			Corner:
     1     2     3     4      6       7
     5    10    13    15     17      20
    25    50    65    75     85     100
   125   250   325   375    425     500
   625  1250  1625  1875   2125    2500
  3125  6250  8125  9375  10625   12500
		

Crossrefs

Cf. A083025, A002144, A002145, A378194.

Programs

Maple

A378193 := proc(n,k)
    option remember;
    local a;
    if k = 0 then
       0;
    else
        for a from procname(n,k-1)+1 do
            if A083025(a) = n-1 then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(seq( A378193(n,d-n),n=1..d-1),d=2..10) ;

Mathematica

u = Map[Map[#[[1]] &, #] &, GatherBy[
    SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 1)/4] && PrimeQ[#] &,
             Flatten[Map[ConstantArray[#[[1]], #[[2]]] &,
             FactorInteger[#]]]], True]} &,
      Range[13000]], #[[2]] &], #[[2]] &]];
r[m_] := Take[u[[m]], 6];
w[m_, n_] := r[m][[n]];
Grid[Table[w[m, n], {m, 1, 6}, {n, 1, 6}]] (* array *)
Table[w[n - k + 1, k], {n, 6}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Nov 19 2024 *)