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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.

%I A378193 #11 Jan 27 2025 05:07:27
%S A378193 1,2,5,3,10,25,4,13,50,125,6,15,65,250,625,7,17,75,325,1250,3125,8,20,
%T A378193 85,375,1625,6250,15625,9,26,100,425,1875,8125,31250,78125,11,29,130,
%U A378193 500,2125,9375,40625,156250,390625,12,30,145,650,2500,10625,46875,203125,781250,1953125
%N 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.
%C A378193 Every positive integer occurs exactly once.
%e A378193 Corner:
%e A378193      1     2     3     4      6       7
%e A378193      5    10    13    15     17      20
%e A378193     25    50    65    75     85     100
%e A378193    125   250   325   375    425     500
%e A378193    625  1250  1625  1875   2125    2500
%e A378193   3125  6250  8125  9375  10625   12500
%p A378193 A378193 := proc(n,k)
%p A378193     option remember;
%p A378193     local a;
%p A378193     if k = 0 then
%p A378193        0;
%p A378193     else
%p A378193         for a from procname(n,k-1)+1 do
%p A378193             if A083025(a) = n-1 then
%p A378193                 return a;
%p A378193             end if;
%p A378193         end do;
%p A378193     end if;
%p A378193 end proc:
%p A378193 seq(seq( A378193(n,d-n),n=1..d-1),d=2..10) ;
%t A378193 u = Map[Map[#[[1]] &, #] &, GatherBy[
%t A378193     SortBy[Map[{#, 1 + Count[Map[IntegerQ[(# - 1)/4] && PrimeQ[#] &,
%t A378193              Flatten[Map[ConstantArray[#[[1]], #[[2]]] &,
%t A378193              FactorInteger[#]]]], True]} &,
%t A378193       Range[13000]], #[[2]] &], #[[2]] &]];
%t A378193 r[m_] := Take[u[[m]], 6];
%t A378193 w[m_, n_] := r[m][[n]];
%t A378193 Grid[Table[w[m, n], {m, 1, 6}, {n, 1, 6}]] (* array *)
%t A378193 Table[w[n - k + 1, k], {n, 6}, {k, n, 1, -1}] // Flatten  (* sequence *)
%t A378193 (* _Peter J. C. Moses_, Nov 19 2024 *)
%Y A378193 Cf. A083025, A002144, A002145, A378194.
%K A378193 nonn,tabl
%O A378193 1,2
%A A378193 _Clark Kimberling_, Jan 14 2025