A191612 Image of A008578 (the noncomposite numbers) under the "forming" transformation.

1, 2, 3, 4, 6, 8, 12, 16, 18, 20, 24, 30, 36, 40, 42, 44, 48, 54, 60, 66, 68, 72, 78, 80, 84, 96, 100, 102, 104, 108, 112, 126, 128, 132, 138, 140, 150, 156, 162, 164, 168, 174, 180, 190, 192, 196, 198, 204, 216, 224, 228

Offset: 1

Author

Jaroslav Krizek, Oct 16 2011

Keywords

• nonn
• easy

Comments

We define a transformation T_f [b(n)] = [c(n)] - the index f means "forming" - of an increasing sequence b(n) of integers b(1), b(2), b(3), ..., b(k) which produces an increasing sequence c(n) of the same length, c(1), c(2), c(3), ..., c(k) such that c(1) = b(1), and for j>1, c(j) is the only integer b(j-1) < c(j) <= b(j), with (b(j)-b(j-1)) | c(j). We say b(n) is forming c(n).

An increasing sequence c(n) is called formed from the increasing sequence b(n) by T_f [b(n)] when there is an increasing sequence b(n) such that b(1) = c(1), for j > 1, b(j) is an integer c(j) <= b(j) < c(j+1) such that difference b(j) - b(j-1) divides c(j).

This transformation T_invf [c(n)] is an inverse of T_f [b(n)], but this inversion of c(n) back to b(n) may not be unique, and there are also increasing sequences c(n) which do not have an image T_invf [c(n)]. We call the latter sequences c(n) "unformed."

Each increasing sequences b(n) can be transform by transformation T_f [b(n)] but this does not apply to transformation T_invf [b(n)]. An increasing sequence c(n) is called totally formed if c(n) = T_f [c(n)] = T_invf [c(n)]. Each totally formed sequence is formed.

There are infinitely many formed, totally formed and unformed increasing sequences.

Examples of totally formed sequences: A047229, A004277, A002808, A000079, A000027.

Examples of formed, but not totally formed, sequences: A000225, A000295, A018252.

Examples of unformed sequences: A000040, A008578, A005117, A005408.

Examples

			a(10) = 20 because 20 is the only integer such that 19 = A008578(9) < 20 <= A008578(10) = 23 and simultaneously is multiple of difference A008578(10) - A008578(9) = 4.
		

Programs

Maple

Tf := proc(L)
        local a,j,c ;
        a := [op(1,L)] ;
        while nops(a) < nops(L)-1 do
                j := nops(a)+1 ;
                for c from op(j-1,L)+1 to op(j,L) do
                        if (c mod ( op(j,L)-op(j-1,L) )) = 0 then
                                a := [op(a),c] ;
                                break;
                        end if;
                end do:
        end do:
        a ;
end proc:
nonc := [seq(A008578(n),n=1..80)] ;
Tf(nonc) ; # R. J. Mathar, Oct 27 2011

Formula

For n > 3, a(n) = A113709(n-2).