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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.

%I A378194 #11 Jan 28 2025 07:54:01
%S A378194 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,
%T A378194 189,486,729,16,15,42,108,297,567,1458,2187,17,19,45,126,324,891,1701,
%U A378194 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
%N 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.
%C A378194 Every positive integer occurs exactly once.
%e A378194 Corner:
%e A378194       1     2     4     5     8     10     13     16      17
%e A378194       3     6     7    11    12     14     15     19      22
%e A378194       9    18    21    33    36     42     45     49      57
%e A378194      27    54    63    99   108    126    135    147     171
%e A378194      81   162   189   297   324    378    405    441     513
%e A378194     243   486   567   891   972   1134   1215   1323    1539
%e A378194     729  1458  1701  2673  2916   3402   3645   3969    4617
%e A378194    2187  4374  5103  8019  8748  10206  10935  11907   13851
%p A378194 A378194 := proc(n, k)
%p A378194     option remember;
%p A378194     local a;
%p A378194     if k = 0 then
%p A378194         0;
%p A378194     else
%p A378194         for a from procname(n, k-1)+1 do
%p A378194             if A065339(a) = n-1 then
%p A378194                 return a;
%p A378194             end if;
%p A378194         end do;
%p A378194     end if;
%p A378194 end proc:
%p A378194 seq(seq( A378194(n, d-n), n=1..d-1), d=2..10) ; # _R. J. Mathar_, Jan 28 2025
%t A378194 u = Map[Map[#[[1]] &, #] &, GatherBy[
%t A378194     SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 3)/4] && PrimeQ[#] &,
%t A378194              Flatten[Map[ConstantArray[#[[1]], #[[2]]] &,
%t A378194              FactorInteger[#]]]], True]} &,
%t A378194       Range[24000]], #[[2]] &], #[[2]] &]];
%t A378194 r[m_] := Take[u[[m]], 10];
%t A378194 w[m_, n_] := r[m][[n]];
%t A378194 Grid[Table[w[m, n], {m, 1, 8}, {n, 1, 9}]]   (* array *)
%t A378194 Table[w[n - k + 1, k], {n, 8}, {k, n, 1, -1}] // Flatten  (* sequence *)
%t A378194 (* _Peter J. C. Moses_, Nov 19 2024 *)
%Y A378194 Cf. A065339, A002144, A002145, A376961, A378193, A072437 (row 1), A000244 (column 0), A025192 (column 1).
%K A378194 nonn,tabl
%O A378194 1,2
%A A378194 _Clark Kimberling_, Jan 14 2025
%E A378194 Definition corrected. - _R. J. Mathar_, Jan 28 2025