A114538 -id:A114538 - cp's OEIS frontend

A191426 Dispersion of (3+[n*r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

1, 4, 2, 9, 6, 3, 17, 12, 7, 5, 30, 22, 14, 11, 8, 51, 38, 25, 20, 15, 10, 85, 64, 43, 35, 27, 19, 13, 140, 106, 72, 59, 46, 33, 24, 16, 229, 174, 119, 98, 77, 56, 41, 28, 18, 373, 284, 195, 161, 127, 93, 69, 48, 32, 21, 606, 462, 318, 263, 208, 153, 114, 80, 54, 36, 23, 983, 750, 517, 428, 339, 250, 187, 132, 90, 61, 40, 26

Offset: 1

Clark Kimberling, Jun 02 2011

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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022342 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...4...9...17..30
  2...6...12..22..38
  3...7...14..25..43
  5...11..20..35..59
  8...15..27..46..77

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

Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3, Fall 1998, p. 176. Solution published in Vol. 12, No. 1, Winter 2000, pp. 61-62.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 2.

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=#rows of T, r1=#rows to show *)
c = 40; c1 = 12; (* c=#cols of T, c1=#cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + 3]
mex[list_] :=  NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191426 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]  (* A191426 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A114537 Dispersion of the primes (an array read by downward antidiagonals).

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 17, 13, 8, 11, 59, 41, 19, 9, 31, 277, 179, 67, 23, 10, 127, 1787, 1063, 331, 83, 29, 12, 709, 15299, 8527, 2221, 431, 109, 37, 14, 5381, 167449, 87803, 19577, 3001, 599, 157, 43, 15, 52711, 2269733, 1128889, 219613, 27457, 4397, 919, 191, 47

Offset: 1

Clark Kimberling, Dec 07 2005

A number is prime if and only if it does not lie in Column 1. As a sequence, a permutation of the natural numbers. The fractal sequence of this dispersion is A022447 and the transposition sequence is A114538.

The dispersion of the composite numbers is given at A114577.

			Northwest corner of the Primeness array:
1   2   3    5    11     31     127       709       5381       52711        648391
4   7  17   59   277   1787   15299    167449    2269733    37139213     718064159
6  13  41  179  1063   8527   87803   1128889   17624813   326851121    7069067389
8  19  67  331  2221  19577  219613   3042161   50728129   997525853   22742734291
9  23  83  431  3001  27457  318211   4535189   77557187  1559861749   36294260117
10  29 109  599  4397  42043  506683   7474967  131807699  2724711961   64988430769
12  37 157  919  7193  72727  919913  14161729  259336153  5545806481  136395369829
14  43 191 1153  9319  96797 1254739  19734581  368345293  8012791231  200147986693
15  47 211 1297 10631 112129 1471343  23391799  440817757  9672485827  243504973489
16  53 241 1523 12763 137077 1828669  29499439  563167303 12501968177  318083817907
18  61 283 1847 15823 173867 2364361  38790341  751783477 16917026909  435748987787
20  71 353 2381 21179 239489 3338989  56011909 1107276647 25366202179  664090238153
21  73 367 2477 22093 250751 3509299  59053067 1170710369 26887732891  705555301183
22  79 401 2749 24859 285191 4030889  68425619 1367161723 31621854169  835122557939
24  89 461 3259 30133 352007 5054303  87019979 1760768239 41192432219 1099216100167
25  97 509 3637 33967 401519 5823667 101146501 2062666783 48596930311 1305164025929
26 101 547 3943 37217 443419 6478961 113256643 2323114841 55022031709 1484830174901
27 103 563 4091 38833 464939 6816631 119535373 2458721501 58379844161 1579041544637

Alexandrov, Lubomir. "On the nonasymptotic prime number distribution." arXiv preprint math/9811096 (1998). (See Appendix.)
Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria, 45 (1997) 157-168.

Robert G. Wilson v, Table of n, a(n) for n = 1..172
Neil Fernandez, An order of primeness, F(p)
Neil Fernandez, An order of primeness [cached copy, included with permission of the author]
Neil Fernandez, The Exploring Primeness Project
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Robert G. Wilson v, The Northwest Corner of the Primeness Array (24 x 24).

Columns 1-13: A018252, A007821, A049078, A049079, A049080, A049081, A058322, A058324, A058325, A058326, A058327, A058328, A093046.
Rows 1-7: A007097, A057450, A057451, A057452, A057453, A057456, A057457.
Diagonal: A181441.
Cf. A000040, A007821, A114538, A006450.
If the antidiagonals are read in the opposite direction we get A138947.

A114537 := proc(r,c) option remember; if c = 1 then A018252(r) ; else ithprime(procname(r,c-1)) ; end if; end proc: # R. J. Mathar, Oct 22 2010

NonPrime[n_] := FixedPoint[n + PrimePi@# + 1 &, n]; t[n_, k_] := Nest[Prime, NonPrime[n], k]; Table[ t[n - k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten
(* or to view the table *) Table[t[n, k], {n, 0, 6}, {k, 0, 10}] // TableForm (* Robert G. Wilson v, Dec 26 2005 *)

T(r,1) = A018252(r). T(r,c) = prime(T(r,c-1)), c>1. [R. J. Mathar, Oct 22 2010]

A191455 Dispersion of (floor(n*e)), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 13, 21, 10, 6, 35, 57, 27, 16, 7, 95, 154, 73, 43, 19, 9, 258, 418, 198, 116, 51, 24, 11, 701, 1136, 538, 315, 138, 65, 29, 12, 1905, 3087, 1462, 856, 375, 176, 78, 32, 14, 5178, 8391, 3974, 2326, 1019, 478, 212, 86, 38, 15, 14075, 22809

Offset: 1

Clark Kimberling, Jun 05 2011

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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...2....5....13...35
  3...8....21...57...154
  4...10...27...73...198
  6...16...43...116..315
  7...19...51...138..375

Cf. A114537, A035513, A035506.

A191455 := proc(r, c)
    option remember;
    if c = 1 then
        A054385(r) ;
    else
        A022843(procname(r, c-1)) ;
    end if;
end proc: # R. J. Mathar, Jan 25 2015

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=Floor[n*E]   (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191455 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191455 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191452 Dispersion of (4,8,12,16,...), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 16, 8, 3, 64, 32, 12, 5, 256, 128, 48, 20, 6, 1024, 512, 192, 80, 24, 7, 4096, 2048, 768, 320, 96, 28, 9, 16384, 8192, 3072, 1280, 384, 112, 36, 10, 65536, 32768, 12288, 5120, 1536, 448, 144, 40, 11, 262144, 131072, 49152, 20480, 6144, 1792, 576

Offset: 1

Clark Kimberling, Jun 05 2011

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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...4....16...64....256
  2...8....32...128...512
  3...12...48...192...768
  5...20...80...320...1280
  6...24...96...384...1536

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A114537, A035513, A035506, A191449.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=4n  (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191452 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191452 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191449 Dispersion of (3,6,9,12,15,...), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 9, 6, 4, 27, 18, 12, 5, 81, 54, 36, 15, 7, 243, 162, 108, 45, 21, 8, 729, 486, 324, 135, 63, 24, 10, 2187, 1458, 972, 405, 189, 72, 30, 11, 6561, 4374, 2916, 1215, 567, 216, 90, 33, 13, 19683, 13122, 8748, 3645, 1701, 648, 270, 99, 39, 14, 59049

Offset: 1

Clark Kimberling, Jun 05 2011

Transpose of A141396.
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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...3....9....27...81
  2...6....18...54...162
  4...12...36...108..324
  5...15...45...135..405
  7...21...63...189..567

Cf. A114537, A035513, A035506,
A054582: dispersion of (2,4,6,8,...).
A191450: dispersion of (2,5,8,11,...).
A191451: dispersion of (4,7,10,13,...).
A191452: dispersion of (4,8,12,16,...).

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=3n (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191449 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191449 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

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

A191448 Dispersion of the odd integers greater than 1, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 15, 11, 9, 6, 31, 23, 19, 13, 8, 63, 47, 39, 27, 17, 10, 127, 95, 79, 55, 35, 21, 12, 255, 191, 159, 111, 71, 43, 25, 14, 511, 383, 319, 223, 143, 87, 51, 29, 16, 1023, 767, 639, 447, 287, 175, 103, 59, 33, 18, 2047, 1535, 1279, 895, 575

Offset: 1

Clark Kimberling, Jun 05 2011

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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...3...7...15..31
  2...5...11..23..47
  4...9...19..39..79
  6...13..27..55..111
  8...17..35..71..143

Ivan Neretin, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals, flattened)

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;
f[n_] :=2n+1 (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191448 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191448 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191545 Dispersion of (floor(9*n/4)), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 9, 13, 11, 7, 20, 29, 24, 15, 8, 45, 65, 54, 33, 18, 10, 101, 146, 121, 74, 40, 22, 12, 227, 328, 272, 166, 90, 49, 27, 14, 510, 738, 612, 373, 202, 110, 60, 31, 16, 1147, 1660, 1377, 839, 454, 247, 135, 69, 36, 17, 2580, 3735, 3098, 1887

Offset: 1

Clark Kimberling, Jun 09 2011

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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1   2    4    9    20
  3   6    13   29   65
  5   11   25   54   121
  7   15   33   74   166
  8   18   40   90   202

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the complement of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] := Floor[9n/4]   (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191545 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191545 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191438 Dispersion of ([n*sqrt(2)+n]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 9, 16, 12, 6, 21, 38, 28, 14, 8, 50, 91, 67, 33, 19, 10, 120, 219, 161, 79, 45, 24, 11, 289, 528, 388, 190, 108, 57, 26, 13, 697, 1274, 936, 458, 260, 137, 62, 31, 15, 1682, 3075, 2259, 1105, 627, 330, 149, 74, 36, 17, 4060, 7423, 5453

Offset: 1

Clark Kimberling, Jun 04 2011

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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1....2....4....9....21
  3....7....16...38...91
  5....12...28...67...161
  6....14...33...79...190
  8....19...45...108..260

Cf. A114537, A035513, A035506, A191438.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[2];
f[n_] := Floor[n*x+n] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191438 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191438 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191536 Dispersion of (3+floor(n*sqrt(2))), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 8, 5, 3, 14, 10, 7, 6, 22, 17, 12, 11, 9, 34, 27, 19, 18, 15, 13, 51, 41, 29, 28, 24, 21, 16, 75, 60, 44, 42, 36, 32, 25, 20, 109, 87, 65, 62, 53, 48, 38, 31, 23, 157, 126, 94, 90, 77, 70, 56, 46, 35, 26, 225, 181, 135, 130, 111, 101, 82, 68, 52, 39

Offset: 1

Clark Kimberling, Jun 06 2011

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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455 and A191536-A191545.

			Northwest corner:
  1...4....8....14...22
  2...5....10...17...27
  3...7....12...19...29
  6...11...18...28...42
  9...15...24...36...54

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=3+Floor[n*Sqrt[2]]   (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]
(* A191536 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191536 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191442 Dispersion of ([n*sqrt(3)+1/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 5, 12, 10, 8, 9, 21, 17, 14, 11, 16, 36, 29, 24, 19, 13, 28, 62, 50, 42, 33, 23, 15, 48, 107, 87, 73, 57, 40, 26, 18, 83, 185, 151, 126, 99, 69, 45, 31, 20, 144, 320, 262, 218, 171, 120, 78, 54, 35, 22, 249, 554, 454, 378, 296, 208, 135, 94

Offset: 1

Clark Kimberling, Jun 04 2011

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:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1....2....3....5....9
  4....7....12...21...36
  6....10...17...29...50
  8....14...24...42...73
  11...19...33...57...99

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[3];
f[n_] := Floor[n*x+1/2] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191442 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191442 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

cp's OEIS Frontend

A191426 Dispersion of (3+[n*r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

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A114537 Dispersion of the primes (an array read by downward antidiagonals).

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Maple

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A191455 Dispersion of (floor(n*e)), by antidiagonals.

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Maple

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A191452 Dispersion of (4,8,12,16,...), by antidiagonals.

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A191449 Dispersion of (3,6,9,12,15,...), by antidiagonals.

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A191448 Dispersion of the odd integers greater than 1, by antidiagonals.

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A191545 Dispersion of (floor(9*n/4)), by antidiagonals.

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A191438 Dispersion of ([n*sqrt(2)+n]), where [ ]=floor, by antidiagonals.

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