A114577 Dispersion of the composite numbers.

1, 4, 2, 9, 6, 3, 16, 12, 8, 5, 26, 21, 15, 10, 7, 39, 33, 25, 18, 14, 11, 56, 49, 38, 28, 24, 20, 13, 78, 69, 55, 42, 36, 32, 22, 17, 106, 94, 77, 60, 52, 48, 34, 27, 19, 141, 125, 105, 84, 74, 68, 50, 40, 30, 23, 184, 164, 140, 115, 100, 93, 70, 57, 45, 35, 29, 236, 212, 183

Offset

1,2

Comments

Column 1 consists of 1 and the primes. As a sequence, this is a permutation of the positive integers. As an array, its fractal sequence is A022446 and its transposition sequence is A114578.

The dispersion of the primes is given at A114537.

References

Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168.

Links

Ivan Neretin, Table of n, a(n) for n = 1..4950

Clark Kimberling, Interspersions and Dispersions.

Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.

Example

Northwest corner:
1   4   9   16  26  39  56   78
2   6   12  21  33  49  69   94
3   8   15  25  38  55  77   105
5   10  18  28  42  60  84   115
7   14  24  36  52  74  100  133
11  20  32  48  68  93  124  162

Mathematica

(* Program computes dispersion array T of increasing sequence s[n] and the fractal sequence f of T; here, T = dispersion of the composite numbers, A114577 *)
r = 40; r1 = 10; (* r = # rows of T, r1 = # rows to show*);
c = 40; c1 = 12; (* c = # cols of T, c1 = # cols to show*);
comp = Select[Range[2, 100000], ! PrimeQ[#] &];
s[n_] := s[n] = comp[[n]]; mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]; rows = {NestList[s, 1, c]}; Do[rows = Append[rows, NestList[s, mex[Flatten[rows]], r]], {r}]; t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]] (* A114577 array *)
u = Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A114577 sequence *)
row[i_] := row[i] = Table[t[i, j], {j, 1, c}];
f[n_] := Select[Range[r], MemberQ[row[#], n] &]
v = Flatten[Table[f[n], {n, 1, 100}]]  (* A022446, fractal sequence *)
(* - Clark Kimberling, Oct 09 2014 *)

Crossrefs

Cf. A022446, A114537, A114578.

Sequence in context: A082156 A283941 A285090 * A285089 A191426 A182702

Adjacent sequences: A114574 A114575 A114576 * A114578 A114579 A114580

Keyword

nonn,tabl

Author

Clark Kimberling, Dec 09 2005

Status

approved