A191449 - OEIS

%I #15 Oct 21 2024 00:51:38

%S 1,3,2,9,6,4,27,18,12,5,81,54,36,15,7,243,162,108,45,21,8,729,486,324,

%T 135,63,24,10,2187,1458,972,405,189,72,30,11,6561,4374,2916,1215,567,

%U 216,90,33,13,19683,13122,8748,3645,1701,648,270,99,39,14,59049

%N Dispersion of (3,6,9,12,15,...), by antidiagonals.

%C Transpose of A141396.

%C Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:

%C (1) s=A000040 (the primes), D=A114537, u=A114538.

%C (2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.

%C (3) s=A007067, D=A035506 (Stolarsky array), u=A133299.

%C More recent examples of dispersions: A191426-A191455.

%F T(i,j)=T(i,1)*T(1,j)=floor((3i-1)/2)*3^(j-1).

%e Northwest corner:

%e   1...3....9....27...81

%e   2...6....18...54...162

%e   4...12...36...108..324

%e   5...15...45...135..405

%e   7...21...63...189..567

%t (* Program generates the dispersion array T of increasing sequence f[n] *)

%t r=40; r1=12; c=40; c1=12;

%t f[n_] :=3n (* complement of column 1 *)

%t mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]

%t rows = {NestList[f, 1, c]};

%t Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];

%t t[i_, j_] := rows[[i, j]];

%t TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]

%t (* A191449 array *)

%t Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191449 sequence *)

%t (* Program by _Peter J. C. Moses_, Jun 01 2011 *)

%Y Cf. A114537, A035513, A035506,

%Y A054582: dispersion of (2,4,6,8,...).

%Y A191450: dispersion of (2,5,8,11,...).

%Y A191451: dispersion of (4,7,10,13,...).

%Y A191452: dispersion of (4,8,12,16,...).

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jun 05 2011