A114578 -id:A114578 - cp's OEIS frontend

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

Clark Kimberling, Dec 09 2005

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.

			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

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

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.

Cf. A022446, A114537, A114578.

(* 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 *)

A114579 Transposition sequence of the Wythoff array.

Original entry on oeis.org

1, 4, 6, 2, 9, 3, 7, 12, 5, 11, 10, 8, 14, 13, 18, 16, 21, 15, 34, 29, 17, 55, 47, 26, 89, 24, 144, 76, 20, 233, 123, 42, 377, 19, 610, 199, 68, 987, 39, 1597, 322, 32, 2584, 521, 110, 4181, 23, 6765, 843, 178, 10946, 63, 17711, 1364, 22, 28657, 2207, 288, 46368, 102, 75025, 3571, 52, 121393, 5778, 466, 196418, 37, 317811, 9349, 754, 514229, 165, 832040, 15127, 28, 1346269, 24476, 1220, 2178309, 267, 3524578, 39603, 84, 5702887, 64079, 1974, 9227465, 25, 14930352, 103682, 3194, 24157817, 432, 39088169, 167761, 136, 63245986, 271443, 5168

Offset: 1

Clark Kimberling, Dec 09 2005

nonn

A self-inverse permutation of the positive integers. Let s(n)=n-1+Floor(n*tau) and F(n)=n-th Fibonacci number. Then F(n+1) is in position s(n) and s(n) is in position F(n+1).

			Start with the northwest corner of the Wythoff array T (A035513):
1 2 3 5 8
4 7 11 18 29
6 10 16 26 42
9 15 24 39 63
a(1)=1 because 1=T(1,1) and T(1,1)=1.
a(2)=4 because 2=T(1,2) and T(2,1)=4.
a(3)=6 because 3=T(1,3) and T(3,1)=6.
a(15)=18 because 15=T(4,2) and T(2,4)=18.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Cf. A035513, A114538, A114578.

Suppose (as at A114538) that T is a rectangular array consisting of all the positive integers, each exactly once. The transposition sequence of T is obtained by placing T(i, j) in position T(j, i) for all i and j.

cp's OEIS Frontend

A114577 Dispersion of the composite numbers.

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