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

4, 6, 9, 8, 12, 16, 10, 15, 21, 26, 14, 18, 25, 33, 39, 20, 24, 28, 38, 49, 56, 22, 32, 36, 42, 55, 69, 78, 27, 34, 48, 52, 60, 77, 94, 106, 30, 40, 50, 68, 74, 84, 105, 125, 141, 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

Offset: 1

Author

R. J. Mathar, Jan 28 2014

Keywords

• nonn
• easy
• tabl

Comments

Row n contains the composites A002808(j) for which A059981(j) = n.

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,...

The 2nd row contains the composites with an index in the 1st row.

Recursively the followup rows contain the composites that need a higher number of applications of A002808 to reach a nonprime.

Examples

			The array starts:
  4,  6,  8, 10, 14, 20, 22, 27, 30, 35,...
  9, 12, 15, 18, 24, 32, 34, 40, 45, 51,...
 16, 21, 25, 28, 36, 48, 50, 57, 64, 72,...
 26, 33, 38, 42, 52, 68, 70, 80, 88, 98,...
 39, 49, 55, 60, 74, 93, 95,110,119,130,...
 56, 69, 77, 84,100,124,126,145,156,170,...
 78, 94,105,115,133,162,165,188,203,218,...
106,125,140,152,174,209,213,242,259,278,...
141,164,183,198,222,266,272,305,326,348,...
		

Crossrefs

Cf. A006508 (column 1), A022449 (row 1), A135044, A236542, A002808.

Programs

Maple

A236536 := proc(n,k)
    option remember ;
    if n = 1 then
        A022449(k) ;
    else
        A002808(procname(n-1,k)) ;
    end if:
end proc:
for d from 2 to 10 do
     for k from d-1 to  by -1 do
        printf("%3d,",A236536(d-k,k)) ;
     end do:
end do:

Mathematica

Composite[n_] := FixedPoint[n + PrimePi[#] + 1&, n + PrimePi[n] + 1];
T[n_, k_] := T[n, k] = If[n == 1, Composite[If[k == 1, 1, Prime[k - 1]]], Composite[T[n - 1, k]]];
Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 16 2023 *)

Formula

T(1,k) = A022449(k).

T(n,k) = A002808( T(n-1,k) ), n>1 .