A191655 -id:A191655 - cp's OEIS frontend

A191741 Dispersion of A047217, (numbers >1 and congruent to 0 or 1 or 2 mod 5), by antidiagonals.

1, 2, 3, 5, 6, 4, 10, 11, 7, 8, 17, 20, 12, 15, 9, 30, 35, 21, 26, 16, 13, 51, 60, 36, 45, 27, 22, 14, 86, 101, 61, 76, 46, 37, 25, 18, 145, 170, 102, 127, 77, 62, 42, 31, 19, 242, 285, 171, 212, 130, 105, 71, 52, 32, 23, 405, 476, 286, 355, 217, 176, 120

Offset: 1

Clark Kimberling, Jun 14 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3, mod 4, or mod 5, see A191655, A191663, A191667, A191702.
...
Suppose that {2,3,4,5,6} is partitioned as {x1, x2} and {x3,x4,x5}.  Let S be the increasing sequence of numbers >1 and congruent to x1 or x2 mod 5, and let T be the increasing sequence of numbers >1 and congruent to x3 or x4 or x5 mod 5.  There are 10 sequences in S, each matched by a (nearly) complementary sequence in T.  Each of the 20 sequences generates a dispersion, as listed here:
...
A191722=dispersion of A008851 (0, 1 mod 5 and >1)
A191723=dispersion of A047215 (0, 2 mod 5 and >1)
A191724=dispersion of A047218 (0, 3 mod 5 and >1)
A191725=dispersion of A047208 (0, 4 mod 5 and >1)
A191726=dispersion of A047216 (1, 2 mod 5 and >1)
A191727=dispersion of A047219 (1, 3 mod 5 and >1)
A191728=dispersion of A047209 (1, 4 mod 5 and >1)
A191729=dispersion of A047221 (2, 3 mod 5 and >1)
A191730=dispersion of A047211 (2, 4 mod 5 and >1)
A191731=dispersion of A047204 (3, 4 mod 5 and >1)
...
A191732=dispersion of A047202 (2,3,4 mod 5 and >1)
A191733=dispersion of A047206 (1,3,4 mod 5 and >1)
A191734=dispersion of A032793 (1,2,4 mod 5 and >1)
A191735=dispersion of A047223 (1,2,3 mod 5 and >1)
A191736=dispersion of A047205 (0,3,4 mod 5 and >1)
A191737=dispersion of A047212 (0,2,4 mod 5 and >1)
A191738=dispersion of A047222 (0,2,3 mod 5 and >1)
A191739=dispersion of A008854 (0,1,4 mod 5 and >1)
A191740=dispersion of A047220 (0,1,3 mod 5 and >1)
A191741=dispersion of A047217 (0,1,2 mod 5 and >1)
...
For further information about these 20 dispersions, see A191722.
...
Regarding the dispersions A191722-A191741, there are general formulas for sequences of the type "(a or b mod m)" and "(a or b or c mod m)" used in the relevant Mathematica programs.

			Northwest corner:
1....2....5....10...17
3....6....11...20...35
4....7....12...21...36
8....15...26...45...76
9....16...27...46...77
13...22...37...62...105

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

Cf. A047204, A047217, A191722, A191731, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=5; c2=6; m[n_]:=If[Mod[n,3]==0,1,0];
f[n_]:=a*m[n+2]+b*m[n+1]+c2*m[n]+5*Floor[(n-1)/3]
Table[f[n], {n, 1, 30}]  (* A047217 *)
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}]] (* A191741 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191741  *)

A083044 Square table read by antidiagonals forms a permutation of the natural numbers: T(n,0) = floor(nx/(x-1))+1, T(n,k+1) = ceiling(xT(n,k)), where x=3/2, n >= 0, k >= 0.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 5, 9, 11, 10, 8, 14, 17, 15, 13, 12, 21, 26, 23, 20, 16, 18, 32, 39, 35, 30, 24, 19, 27, 48, 59, 53, 45, 36, 29, 22, 41, 72, 89, 80, 68, 54, 44, 33, 25, 62, 108, 134, 120, 102, 81, 66, 50, 38, 28, 93, 162, 201, 180, 153, 122, 99, 75, 57, 42, 31, 140, 243

Offset: 0

Paul D. Hanna, Apr 18 2003

First row is A061419, first column is T(n,0) = A016777(n) = 3n+1 (namely the numbers not of the form ceiling(3*k/2) for any natural number k, in increasing order), main diagonal is A083045, antidiagonal sums give A083046. [further detail on first column added by Glen Whitney, Aug 03 2018]
A083044 is the dispersion of the sequence A007494 of positive integers congruent to (0 or 2) mod 3; see A191655. - Clark Kimberling, Jun 10 2011
If T(n+1,k) - T(n,k) = 2m, then T(n+1,k+1) - T(n,k+1) = ceiling(3T(n+1,k)/2) - ceiling(3T(n,k)/2) = ceiling(3T(n,k)/2 + 3m) - ceiling(3T(n,k)/2) = 3m. Similarly, if T(n+1,k) - T(n,k) = 2m+1, then T(n+1,k+1) - T(n,k+1) = ceiling(3T(n,k)/2 + 3m + 3/2) - ceiling(3T(n,k)/2) = {3m+1 or 3m+2, according to whether T(n,k) is even or odd}. The first differences of the first column T(n,0) are periodic: (3)*. The parities of the first column T(n,0) are periodic: (odd,even)*. Hence by induction using the prior two observations, the first differences and parities of every column will be periodic; e.g., for the second column T(n,2): the first differences are (4,5)* and the parities are (even,even,odd,odd)*; for the third column T(n,3): (6,8,6,7)* and (odd,odd,odd,odd,even,even,even,even)*; for the fourth column T(n,4): (9,12,9,10,9,12,9,11)* and (o,e,e,o,o,e,e,o,e,o,o,e,e,o,o,e)*. Is the period length of the first differences of column k always 2^{k-1}? And is the period length of parities always 2^k? Does every integer > 2 occur as T(n+1,k) - T(n,k) for some n and k? Is the smallest first difference in column k always A061418(k+1)? And is the largest first difference in column k always A061419(k+2)? - Glen Whitney, Aug 03 2018
Consider the following two-player game: Start with two nonempty piles of counters. Players alternate taking turns consisting of first discarding one of the piles and then dividing the remaining pile into two nonempty piles. The smaller pile may always be discarded; the larger pile may only be discarded if the smaller pile is at least half as large. (Either pile may be discarded if they are equal.) The player who cannot move (because the configuration has reached two piles of one counter each) loses. Then the numbers c for which two piles of size c is a losing configuration (for the player whose turn it is) are exactly T(4,k) for k > 1, together with 1,3,5, and 9. - Glen Whitney, Aug 03 2018

			Table begins:
   1  2  3   5   8  12  18  27  41   62   93  140 ...
   4  6  9  14  21  32  48  72 108  162  243  365 ...
   7 11 17  26  39  59  89 134 201  302  453  680 ...
  10 15 23  35  53  80 120 180 270  405  608  912 ...
  13 20 30  45  68 102 153 230 345  518  777 1166 ...
  16 24 36  54  81 122 183 275 413  620  930 1395 ...
  19 29 44  66  99 149 224 336 504  756 1134 1701 ...
  22 33 50  75 113 170 255 383 575  863 1295 1943 ...
  25 38 57  86 129 194 291 437 656  984 1476 2214 ...
  28 42 63  95 143 215 323 485 728 1092 1638 2457 ...
  31 47 71 107 161 242 363 545 818 1227 1841 2762 ...

Cf. A061419, A083045, A083046, A083047, A083050.
Row in which a number occurs: A163491.
Column in which a number occurs: A087088.

T(A163491(n)-1, A087088(n)-1) = n. - Peter Munn, Jul 16 2020 [corrected Peter Munn, Aug 02 2020]

A191703 Dispersion of A016861, (5k+1), by antidiagonals.

Original entry on oeis.org

1, 6, 2, 31, 11, 3, 156, 56, 16, 4, 781, 281, 81, 21, 5, 3906, 1406, 406, 106, 26, 7, 19531, 7031, 2031, 531, 131, 36, 8, 97656, 35156, 10156, 2656, 656, 181, 41, 9, 488281, 175781, 50781, 13281, 3281, 906, 206, 46, 10, 2441406, 878906, 253906, 66406, 16406

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1...6... 31....156...781
2...11...56....281...1406
3...16...81....406...2031
4...21...106...531...2656
5...26...131...656...3281
7...36...181...906...4531

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

Cf. A047202, A016861, A191708, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n+1
Table[f[n], {n, 1, 30}]  (* A016861 *)
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}]] (* A191703 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191703 *)

A191704 Dispersion of A016873, (5k+2), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 7, 12, 4, 32, 57, 17, 5, 157, 282, 82, 22, 6, 782, 1407, 407, 107, 27, 8, 3907, 7032, 2032, 532, 132, 37, 9, 19532, 35157, 10157, 2657, 657, 182, 42, 10, 97657, 175782, 50782, 13282, 3282, 907, 207, 47, 11, 488282, 878907, 253907, 66407, 16407, 4532

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1...2....7.....32....157
3...12...57....282...1407
4...17...82....407...2032
5...22...107...532...2657
6...27...132...657...3282
6...37...182...907...4532

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

Cf. A047207, A016873, A191709, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n-3
Table[f[n], {n, 1, 30}] (* A016873 *)
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}]] (* A191704 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191704 *)

A191705 Dispersion of A016873, (5k+3), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 13, 8, 4, 63, 38, 18, 5, 313, 188, 88, 23, 6, 1563, 938, 438, 113, 28, 7, 7813, 4688, 2188, 563, 138, 33, 9, 39063, 23438, 10938, 2813, 688, 163, 43, 10, 195313, 117188, 54688, 14063, 3438, 813, 213, 48, 11, 976563, 585938, 273438, 70313, 17188

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1...3....13....63....313
2...8....38....188...938
4...18...88....438...2188
5...23...113...563...2813
6...28...138...688...3438
7...33...163...813...4063

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

Cf. A016885, A032763, A191710, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n-2
Table[f[n], {n, 1, 30}] (* A016885 *)
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}]] (* A191705 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191705 *)

A191706 Dispersion of A016873, (5k+4), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 19, 9, 3, 94, 44, 14, 5, 469, 219, 69, 24, 6, 2344, 1094, 344, 119, 29, 7, 11719, 5469, 1719, 594, 144, 34, 8, 58594, 27344, 8594, 2969, 719, 169, 39, 10, 292969, 136719, 42969, 14844, 3594, 844, 194, 49, 11, 1464844, 683594, 214844, 74219, 17969

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1...4....19....94....469
2...9....44....219...1094
3...14...69....344...1719
5...24...119...594...2969
6...29...144...719...3594
7...34...169...844...4219

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

Cf. A001068, A016897, A191711, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n-1
Table[f[n], {n, 1, 30}] (* A016897 *)
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}]] (* A191706 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191706 *)

A191707 Dispersion of A016873, (numbers >1 and congruent to 1, 2, 3, or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 5, 3, 7, 10, 4, 9, 13, 15, 6, 12, 17, 19, 20, 8, 16, 22, 24, 26, 25, 11, 21, 28, 31, 33, 32, 30, 14, 27, 36, 39, 42, 41, 38, 35, 18, 34, 46, 49, 53, 52, 48, 44, 40, 23, 43, 58, 62, 67, 66, 61, 56, 51, 45, 29, 54, 73, 78, 84, 83, 77, 71, 64, 57, 50, 37

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1....2....3....4....6
5....7....9....12...16
10...13...17...22...28
15...19...24...31...39
20...26...33...42...53
25...32...41...52...66

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

Cf. A047201, A008587, A191702, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=3; c2=4; d=6; m[n_]:=If[Mod[n,4]==0,1,0];
f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
Table[f[n], {n, 1, 30}]  (* A047201 *)
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}]] (* A191707 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191707  *)

A191708 Dispersion of A047202, (numbers >1 and congruent to 0, 2, 3, or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 6, 3, 8, 11, 4, 10, 14, 16, 5, 13, 18, 20, 21, 7, 17, 23, 25, 27, 26, 9, 22, 29, 32, 34, 33, 31, 12, 28, 37, 40, 43, 42, 39, 36, 15, 35, 47, 50, 54, 53, 49, 45, 41, 19, 44, 59, 63, 68, 67, 62, 57, 52, 46, 24, 55, 74, 79, 85, 84, 78, 72, 65, 58, 51, 30

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1....2....3....4....5
6....8....10...13...17
11...14...18...23...29
16...20...25...32...40
21...27...34...43...54
26...33...42...53...67

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

Cf. A047202, A016861, A191703, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=3; c2=4; d=5; m[n_]:=If[Mod[n,4]==0,1,0];
f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
Table[f[n], {n, 1, 30}]  (* A047202 *)
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}]] (* A191708 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191708  *)

A191709 Dispersion of A047202, (numbers >1 and congruent to 0, 1, 3, or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 8, 6, 10, 12, 11, 9, 14, 16, 17, 15, 13, 19, 21, 23, 22, 20, 18, 25, 28, 30, 29, 27, 26, 24, 33, 36, 39, 38, 35, 32, 34, 31, 43, 46, 50, 49, 45, 41, 37, 44, 40, 55, 59, 64, 63, 58, 53, 48, 42, 56, 51, 70, 75, 81, 80, 74, 68, 61, 54, 47, 71

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1....3....5....8....11
2....4....6....9....13
7....10...14...19...25
12...16...21...28...36
17...23...30...39...50
22...29...38...49...63

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

Cf. A047207, A016873, A191704, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=4; c2=5; d=6; m[n_]:=If[Mod[n,4]==0,1,0];
f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
Table[f[n], {n, 1, 30}]  (* A047207 *)
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}]] (* A191709 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191709  *)

A191710 Dispersion of A032763, (numbers >1 and congruent to 0, 1, 2, or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 6, 7, 11, 13, 9, 10, 15, 17, 18, 12, 14, 20, 22, 24, 23, 16, 19, 26, 29, 31, 30, 28, 21, 25, 34, 37, 40, 39, 36, 33, 27, 32, 44, 47, 51, 50, 46, 42, 38, 35, 41, 56, 60, 65, 64, 59, 54, 49, 43, 45, 52, 71, 76, 82, 81, 75, 69, 62, 55, 48, 57

Offset: 1

Clark Kimberling, Jun 12 2011

For a background discussion of dispersions and their fractal sequences, see A191426.  For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion.  Each complement (beginning with its first term >1) also generates a dispersion.  The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

			Northwest corner:
1....2....4....6....9
3....5....7....10...14
8....11...15...20...26
13...17...22...29...37
18...24...31...40...51
23...30...39...50...64

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

Cf. A032763, A016885, A191705, A191426.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=4; c2=5; d=6; m[n_]:=If[Mod[n,4]==0,1,0];
f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
Table[f[n], {n, 1, 30}]  (* A032763 *)
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}]] (* A191710 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191710  *)

cp's OEIS Frontend

A191741 Dispersion of A047217, (numbers >1 and congruent to 0 or 1 or 2 mod 5), by antidiagonals.

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A083044 Square table read by antidiagonals forms a permutation of the natural numbers: T(n,0) = floor(n*x/(x-1))+1, T(n,k+1) = ceiling(x*T(n,k)), where x=3/2, n >= 0, k >= 0.

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A191703 Dispersion of A016861, (5k+1), by antidiagonals.

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A191704 Dispersion of A016873, (5k+2), by antidiagonals.

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A191705 Dispersion of A016873, (5k+3), by antidiagonals.

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A191706 Dispersion of A016873, (5k+4), by antidiagonals.

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A191707 Dispersion of A016873, (numbers >1 and congruent to 1, 2, 3, or 4 mod 5), by antidiagonals.

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A191708 Dispersion of A047202, (numbers >1 and congruent to 0, 2, 3, or 4 mod 5), by antidiagonals.

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A191709 Dispersion of A047202, (numbers >1 and congruent to 0, 1, 3, or 4 mod 5), by antidiagonals.

A083044 Square table read by antidiagonals forms a permutation of the natural numbers: T(n,0) = floor(nx/(x-1))+1, T(n,k+1) = ceiling(xT(n,k)), where x=3/2, n >= 0, k >= 0.