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

%I A191612 #24 May 10 2019 21:19:05
%S A191612 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,
%T A191612 96,100,102,104,108,112,126,128,132,138,140,150,156,162,164,168,174,
%U A191612 180,190,192,196,198,204,216,224,228
%N A191612 Image of A008578 (the noncomposite numbers) under the "forming" transformation.
%C A191612 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).
%C A191612 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).
%C A191612 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."
%C A191612 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.
%C A191612 There are infinitely many formed, totally formed and unformed increasing sequences.
%C A191612 Examples of totally formed sequences: A047229, A004277, A002808, A000079, A000027.
%C A191612 Examples of formed, but not totally formed, sequences: A000225, A000295, A018252.
%C A191612 Examples of unformed sequences: A000040, A008578, A005117, A005408.
%F A191612 For n > 3, a(n) = A113709(n-2).
%e A191612 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.
%p A191612 Tf := proc(L)
%p A191612         local a,j,c ;
%p A191612         a := [op(1,L)] ;
%p A191612         while nops(a) < nops(L)-1 do
%p A191612                 j := nops(a)+1 ;
%p A191612                 for c from op(j-1,L)+1 to op(j,L) do
%p A191612                         if (c mod ( op(j,L)-op(j-1,L) )) = 0 then
%p A191612                                 a := [op(a),c] ;
%p A191612                                 break;
%p A191612                         end if;
%p A191612                 end do:
%p A191612         end do:
%p A191612         a ;
%p A191612 end proc:
%p A191612 nonc := [seq(A008578(n),n=1..80)] ;
%p A191612 Tf(nonc) ; # _R. J. Mathar_, Oct 27 2011
%K A191612 nonn,easy
%O A191612 1,2
%A A191612 _Jaroslav Krizek_, Oct 16 2011