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A236536 Array T(n,k) read along antidiagonals: the composites of order of compositeness n in row n.

%I A236536 #17 Sep 16 2023 12:02:07
%S A236536 4,6,9,8,12,16,10,15,21,26,14,18,25,33,39,20,24,28,38,49,56,22,32,36,
%T A236536 42,55,69,78,27,34,48,52,60,77,94,106,30,40,50,68,74,84,105,125,141,
%U A236536 35,45,57,70,93,100,115,140,164,184,44,51,64,80,95,124,133,152,183,212,236,46,63,72,88,110,126,162,174,198,235,270,299
%N A236536 Array T(n,k) read along antidiagonals: the composites of order of compositeness n in row n.
%C A236536 Row n contains the composites A002808(j) for which A059981(j) = n.
%C A236536 The 1st row contains the composites with a nonprime index, A002808(1)=4, A002808(2)=6, A002808(3)=8, A002808(5)=10, A002808(7)=14,...
%C A236536 The 2nd row contains the composites with an index in the 1st row.
%C A236536 Recursively the followup rows contain the composites that need a higher number of applications of A002808 to reach a nonprime.
%F A236536 T(1,k) = A022449(k).
%F A236536 T(n,k) = A002808( T(n-1,k) ), n>1 .
%e A236536 The array starts:
%e A236536   4,  6,  8, 10, 14, 20, 22, 27, 30, 35,...
%e A236536   9, 12, 15, 18, 24, 32, 34, 40, 45, 51,...
%e A236536  16, 21, 25, 28, 36, 48, 50, 57, 64, 72,...
%e A236536  26, 33, 38, 42, 52, 68, 70, 80, 88, 98,...
%e A236536  39, 49, 55, 60, 74, 93, 95,110,119,130,...
%e A236536  56, 69, 77, 84,100,124,126,145,156,170,...
%e A236536  78, 94,105,115,133,162,165,188,203,218,...
%e A236536 106,125,140,152,174,209,213,242,259,278,...
%e A236536 141,164,183,198,222,266,272,305,326,348,...
%p A236536 A236536 := proc(n,k)
%p A236536     option remember ;
%p A236536     if n = 1 then
%p A236536         A022449(k) ;
%p A236536     else
%p A236536         A002808(procname(n-1,k)) ;
%p A236536     end if:
%p A236536 end proc:
%p A236536 for d from 2 to 10 do
%p A236536      for k from d-1 to  by -1 do
%p A236536         printf("%3d,",A236536(d-k,k)) ;
%p A236536      end do:
%p A236536 end do:
%t A236536 Composite[n_] := FixedPoint[n + PrimePi[#] + 1&, n + PrimePi[n] + 1];
%t A236536 T[n_, k_] := T[n, k] = If[n == 1, Composite[If[k == 1, 1, Prime[k - 1]]], Composite[T[n - 1, k]]];
%t A236536 Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Sep 16 2023 *)
%Y A236536 Cf. A006508 (column 1), A022449 (row 1), A135044, A236542, A002808.
%K A236536 nonn,easy,tabl
%O A236536 1,1
%A A236536 _R. J. Mathar_, Jan 28 2014