A378194 Rectangular array, read by descending antidiagonals: row n shows the integers m such that the number of primes of the form 4k+3 (including multiplicities) that divide m is n-1.

1, 2, 3, 4, 6, 9, 5, 7, 18, 27, 8, 11, 21, 54, 81, 10, 12, 33, 63, 162, 243, 13, 14, 36, 99, 189, 486, 729, 16, 15, 42, 108, 297, 567, 1458, 2187, 17, 19, 45, 126, 324, 891, 1701, 4374, 6561, 20, 22, 49, 135, 378, 972, 2673, 5103, 13122, 19683, 25, 23, 57, 147, 405, 1134, 2916, 8019, 15309, 39366, 59049, 26, 24, 66, 171, 441, 1215, 3402, 8748, 24057, 45927, 118098, 177147

Offset: 1

Author

Clark Kimberling, Jan 14 2025

Keywords

• nonn
• tabl

Comments

Every positive integer occurs exactly once.

Examples

			Corner:
      1     2     4     5     8     10     13     16      17
      3     6     7    11    12     14     15     19      22
      9    18    21    33    36     42     45     49      57
     27    54    63    99   108    126    135    147     171
     81   162   189   297   324    378    405    441     513
    243   486   567   891   972   1134   1215   1323    1539
    729  1458  1701  2673  2916   3402   3645   3969    4617
   2187  4374  5103  8019  8748  10206  10935  11907   13851
		

Crossrefs

Cf. A065339, A002144, A002145, A376961, A378193, A072437 (row 1), A000244 (column 0), A025192 (column 1).

Programs

Maple

A378194 := proc(n, k)
    option remember;
    local a;
    if k = 0 then
        0;
    else
        for a from procname(n, k-1)+1 do
            if A065339(a) = n-1 then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(seq( A378194(n, d-n), n=1..d-1), d=2..10) ; # R. J. Mathar, Jan 28 2025

Mathematica

u = Map[Map[#[[1]] &, #] &, GatherBy[
    SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 3)/4] && PrimeQ[#] &,
             Flatten[Map[ConstantArray[#[[1]], #[[2]]] &,
             FactorInteger[#]]]], True]} &,
      Range[24000]], #[[2]] &], #[[2]] &]];
r[m_] := Take[u[[m]], 10];
w[m_, n_] := r[m][[n]];
Grid[Table[w[m, n], {m, 1, 8}, {n, 1, 9}]]   (* array *)
Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten  (* sequence *)
(* Peter J. C. Moses, Nov 19 2024 *)

Extensions

Definition corrected. - R. J. Mathar, Jan 28 2025