A008851 -id:A008851 - cp's OEIS frontend

A191722 Dispersion of A008851, (numbers >1 and congruent to 0 or 1 mod 5), by antidiagonals.

1, 5, 2, 15, 6, 3, 40, 16, 10, 4, 101, 41, 26, 11, 7, 255, 105, 66, 30, 20, 8, 640, 265, 166, 76, 51, 21, 9, 1601, 665, 416, 191, 130, 55, 25, 12, 4005, 1665, 1041, 480, 326, 140, 65, 31, 13, 10015, 4165, 2605, 1201, 816, 351, 165, 80, 35, 14, 25040, 10415

Offset: 1

Clark Kimberling, Jun 13 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)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191722 has 1st col A047202, all else A008851
A191723 has 1st col A047206, all else A047215
A191724 has 1st col A032793, all else A047218
A191725 has 1st col A047223, all else A047208
A191726 has 1st col A047205, all else A047216
A191727 has 1st col A047212, all else A047219
A191728 has 1st col A047222, all else A047209
A191729 has 1st col A008854, all else A047221
A191730 has 1st col A047220, all else A047211
A191731 has 1st col A047217, all else A047204
...
A191732 has 1st col A000851, all else A047202
A191733 has 1st col A047215, all else A047206
A191734 has 1st col A047218, all else A032793
A191735 has 1st col A047208, all else A047223
A191736 has 1st col A047216, all else A047205
A191737 has 1st col A047219, all else A047212
A191738 has 1st col A047209, all else A047222
A191739 has 1st col A047221, all else A008854
A191740 has 1st col A047211, all else A047220
A191741 has 1st col A047204, all else A047217
...
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....5....15...40...101
2....6....16...41...105
3....10...26...66...166
4....11...30...76...191
7....20...51...130..326
8....21...55...140..351

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047202, A008851, A191732, A191702, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=5; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A008851 *)
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}]]
(* A191722 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191722  *)

A008548 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+1).

Original entry on oeis.org

1, 1, 6, 66, 1056, 22176, 576576, 17873856, 643458816, 26381811456, 1213563326976, 61891729675776, 3465936861843456, 211422148572450816, 13953861805781753856, 990724188210504523776, 75295038303998343806976, 6098898102623865848365056, 524505236825652462959394816

Offset: 0

Joe Keane (jgk(AT)jgk.org)

a(n), n>=1, enumerates increasing sextic (6-ary) trees with n vertices. - Wolfdieter Lang, Sep 14 2007

Hankel transform is A169620. - Paul Barry, Dec 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300 (first 50 terms from T. D. Noe)
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), Article 10.6.7, Table 6.3.

Cf. A001147, A007559, A007696, A016861, A034687, A034688, A052562, A047055, A049385, A051150.

List([0..20], n-> Product([0..n], k-> 5*k+1)); # G. C. Greubel, Aug 16 2019

[(&*[5*k+1: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 16 2019

a := n -> mul(5*k+1, k=0..n-1);
G(x):=(1-5*x)^(-1/5): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..16); # Zerinvary Lajos, Apr 03 2009
H := hypergeom([1, 1/5], [], 5*x):
seq(coeff(series(H,x,20),x,n),n=0..16); # Peter Luschny, Oct 08 2015

Table[Product[5k+1,{k,0,n-1}],{n,0,20}]  (* Harvey P. Dale, Apr 23 2011 *)
FoldList[Times,1,NestList[#+5&,1,20]] (* Ray Chandler, Apr 23 2011 *)
FoldList[Times,1,5Range[0, 25] + 1] (* Vincenzo Librandi, Jun 10 2013 *)

x='x+O('x^33); Vec(serlaplace((1-5*x)^(-1/5))) \\ Joerg Arndt, Apr 24 2011

vector(20, n, n--; prod(k=0, n-1, 5*k+1)) \\ Altug Alkan, Oct 08 2015

[product(5*k+1 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 16 2019

a(n) = A049385(n, 1) (first column of triangle).
E.g.f.: (1-5*x)^(-1/5).
a(n) ~ 2^(1/2)*Pi^(1/2)*gamma(1/5)^-1*n^(-3/10)*5^n*e^-n*n^n*{1 + 1/300*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = Sum_{k=0..n} (-5)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
G.f.: 1/(1-x/(1-5x/(1-6x/(1-10x/(1-11x/(1-15x/(1-16x/(1-20x/(1-21x/(1-25x/(1-.../(1-A008851(n+1)*x/(1-... (continued fraction). - Paul Barry, Dec 03 2009
a(n)=(-4)^n*Sum_{k=0..n} (5/4)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0) where Q(k) = 1 - x*(5*k+1)/(1 - x*(5*k+5)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
G.f.: G(0)/2, where G(k)= 1  + 1/(1 - (5*k+1)*x/((5*k+1)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
a(n) = (10n-18)*a(n-2) + (5n-6)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 12 2013
Let T(x) = 1/(1 - 4*x)^(1/4) be the e.g.f. for the sequence of triple factorial numbers A007696. Then the e.g.f. A(x) for the quintuple factorial numbers satisfies T( Integral_{t = 0..x} A(t) dt ) = A(x). Cf. A007559 and A007696. - Peter Bala, Jan 02 2015
O.g.f.: hypergeom([1, 1/5], [], 5*x). - Peter Luschny, Oct 08 2015
a(n) = 5^n * Gamma(n + 1/5) / Gamma(1/5). - Artur Jasinski, Aug 23 2016
D-finite with recurrence: a(n) +(-5*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/5^4)^(1/5)*(Gamma(1/5) - Gamma(1/5, 1/5)). - Amiram Eldar, Dec 19 2022

A003226 Automorphic numbers: m^2 ends with m.

Original entry on oeis.org

0, 1, 5, 6, 25, 76, 376, 625, 9376, 90625, 109376, 890625, 2890625, 7109376, 12890625, 87109376, 212890625, 787109376, 1787109376, 8212890625, 18212890625, 81787109376, 918212890625, 9918212890625, 40081787109376, 59918212890625, 259918212890625, 740081787109376

Offset: 1

N. J. A. Sloane

Also called curious numbers.
For entries after the second, two successive terms sum up to a total having the form 10^n + 1. - Lekraj Beedassy, Apr 29 2005 [This comment is clearly wrong as stated. The sums of two consecutive terms are 1, 6, 11, 31, 101, 452, 1001, 10001, 100001, 200001, 1000001, 3781250, .... - T. D. Noe, Nov 14 2010]
If a d-digit number n is in the sequence, then so is 10^d+1-n. However, the same number can be 10^d+1-n for different n in the sequence (e.g., 10^3+1-376 = 10^4+1-9376 = 625), which spoils Beedassy's comment. - Robert Israel, Jun 19 2015
Substring of both its square and its cube not congruent to 0 (mod 10). See A029943. - Robert G. Wilson v, Jul 16 2005
a(n)^k ends with a(n) for k > 0; see also A029943. - Reinhard Zumkeller, Nov 26 2011
Apart from initial term, a subsequence of A046831. - M. F. Hasler, Dec 05 2012
This is also the sequence of numbers such that the n-th m-gonal number ends in n for any m == 0,4,8,16 (mod 20). - Robert Dawson, Jul 09 2018
Apart from 6, a subsequence of A301912. - Robert Dawson, Aug 01 2018

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 76, p. 26, Ellipses, Paris 2008.
V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2 (No. 3, 1969), 170-174.
Jan Gullberg, Mathematics, From the Birth of Numbers, W. W. Norton & Co., NY, page 253-254.
B. A. Naik, 'Automorphic numbers' in 'Science Today'(subsequently renamed '2001') May 1982 pp. 59, Times of India, Mumbai.
Ya. I. Perelman, Algebra can be fun, pp. 97-98.
Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Hoboken, 2005, p. 64.
C. P. Schut, Idempotents. Report AM-R9101, Centrum voor Wiskunde en Informatica, Amsterdam, 1991.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Chai Wah Wu, Table of n, a(n) for n = 1..1786 (terms n = 1..200 from Vincenzo Librandi)
Robert Dawson, On Some Sequences Related to Sums of Powers, J. Int. Seq., Vol. 21 (2018), Article 18.7.6.
R. A. Fairbairn, More on automorphic numbers, J. Rec. Math., 2.3 (1969), 170-174. (Annotated scanned copy)
V. deGuerre & R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1.3 (1968), 173-179
M. Beeler, R. W. Gosper, and R. Schroeppel, HAKMEM. MIT AI Memo 239, Feb 29 1972 (Item 59 provides a 40-digit member of this sequence).
Shyam Sunder Gupta, Elegance of Squares, Cubes, and Higher Powers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 2, 29-81.
W. Schneider, Automorphic Numbers
C. P. Schut, Idempotents, Report AM-R9101, Centre for Mathematics and Computer Science, Amsterdam, 1991. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Automorphic Number
Wikipedia, Automorphic number
Xiaolong Ron Yu, Curious Numbers, Pi Mu Epsilon Journal, Spring 1999, pp. 819-823.
Index entries for sequences related to automorphic numbers

Cf. A008851, A018247, A018248, A018834, A033819, A035383, A046831, A052228.

import Data.List (isSuffixOf)
a003226 n = a003226_list !! (n-1)
a003226_list = filter (\x -> show x `isSuffixOf` show (x^2)) a008851_list
-- Reinhard Zumkeller, Jul 27 2011

[n: n in [0..10^7] | Intseq(n^2)[1..#Intseq(n)] eq Intseq(n)]; // Vincenzo Librandi, Jul 03 2015

V:= proc(m) option remember;
  select(t -> t^2 - t mod 10^m = 0, map(s -> seq(10^(m-1)*j+s, j=0..9), V(m-1)))
end proc:
V(0):= {0,1}:
V(1):= {5,6}:
sort(map(op,[V(0),seq(V(i) minus V(i-1),i=1..50)])); # Robert Israel, Jun 19 2015

f[k_] := (r = Reduce[0 < 10^k < n < 10^(k + 1) && n^2 == m*10^(k + 1) + n, {n, m}, Integers]; If[Head[r] === And, n /. ToRules[r], n /. {ToRules[r]}]); Flatten[ Join[{0, 1}, Table[f[k], {k, 0, 13}]]] (* Jean-François Alcover, Dec 01 2011 *)
Union@ Join[{1}, Array[PowerMod[5, 2^#, 10^#] &, 16, 0], Array[PowerMod[16, 5^#, 10^#] &, 16, 0]] (* Robert G. Wilson v, Jul 23 2018 *)

is_A003226(n)={n<2 || 10^valuation(n^2-n,10)>n} \\ M. F. Hasler, Dec 05 2012

A003226(n)={ n<3 & return(n-1); my(i=10,j=10,b=5,c=6,a=b); for( k=4,n, while(b<=a, b=b^2%i*=10); while(c<=a, c=(2-c)*c%j*=10); a=min(b,c)); a } \\ M. F. Hasler, Dec 06 2012

from itertools import count, islice
from sympy.ntheory.modular import crt
def A003226_gen(): # generator of terms
    a = 0
    yield from (0,1)
    for n in count(0):
        b = sorted((int(crt(m:=(1< a:
            yield from b
            a = b[1]
        elif b[1] > a:
            yield b[1]
            a = b[1]
A003226_list = list(islice(A003226_gen(),15)) # Chai Wah Wu, Jul 25 2022

def automorphic(maxdigits, pow, base=10) :
    morphs = [[0]]
    for i in range(maxdigits):
        T=[d*base^i+x for x in morphs[-1] for d in range(base)]
        morphs.append([x for x in T if x^pow % base^(i+1) == x])
    res = list(set(sum(morphs, []))); res.sort()
    return res
# call with pow=2 for this sequence, Eric M. Schmidt, Feb 09 2014

Equals {0, 1} union A007185 union A016090.

More terms from Michel ten Voorde, Apr 11 2001
Edited by David W. Wilson, Sep 26 2002
Incorrect statement removed from title by Robert Dawson, Jul 09 2018

A191723 Dispersion of A047215, (numbers >1 and congruent to 0 or 2 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 7, 4, 12, 17, 10, 6, 30, 42, 25, 15, 8, 75, 105, 62, 37, 20, 9, 187, 262, 155, 92, 50, 22, 11, 467, 655, 387, 230, 125, 55, 27, 13, 1167, 1637, 967, 575, 312, 137, 67, 32, 14, 2917, 4092, 2417, 1437, 780, 342, 167, 80, 35, 16, 7292, 10230, 6042

Offset: 1

Clark Kimberling, Jun 13 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....12....30
3....7....17...42....105
4....10...25...62....155
6....15...37...92....230
8....20...50...125...312
9....22...55...137...342

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047206, A047215, A191733, A191722, 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; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047215 *)
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}]] (* A191722 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191722  *)

A191724 Dispersion of A047218, (numbers >1 and congruent to 0 or 3 mod 5), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 8, 5, 4, 20, 13, 10, 6, 50, 33, 25, 15, 7, 125, 83, 63, 38, 18, 9, 313, 208, 158, 95, 45, 23, 11, 783, 520, 395, 238, 113, 58, 28, 12, 1958, 1300, 988, 595, 283, 145, 70, 30, 14, 4895, 3250, 2470, 1488, 708, 363, 175, 75, 35, 16, 12238, 8125, 6175

Offset: 1

Clark Kimberling, Jun 13 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....3....8....20....50
2....5....13...33....83
4....10...25...63....158
6....15...38...95....238
7....18...45...113...283
9....23...58...145...363

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A032793, A047218, A191734, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=5; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047218 *)
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}]] (* A191724 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191724  *)

A191725 Dispersion of A047208, (numbers >1 and congruent to 0 or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 10, 5, 3, 25, 14, 9, 6, 64, 35, 24, 15, 7, 160, 89, 60, 39, 19, 8, 400, 224, 150, 99, 49, 20, 11, 1000, 560, 375, 249, 124, 50, 29, 12, 2500, 1400, 939, 624, 310, 125, 74, 30, 13, 6250, 3500, 2349, 1560, 775, 314, 185, 75, 34, 16, 15625, 8750, 5874

Offset: 1

Clark Kimberling, Jun 13 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....4....10....25....64
2....5....14....35...89
3....9....24...60...150
6....15...39...99...249
7....19...49...124..310
8....20...50...125...314

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047223, A047208, A191735, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=4; b=5; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047208 *)
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}]] (* A191725 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191725  *)

A191726 Dispersion of A047216, (numbers >1 and congruent to 1 or 2 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 6, 7, 4, 16, 17, 11, 5, 41, 42, 27, 12, 8, 102, 106, 67, 31, 21, 9, 256, 266, 167, 77, 52, 22, 10, 641, 666, 417, 192, 131, 56, 26, 13, 1602, 1666, 1042, 481, 327, 141, 66, 32, 14, 4006, 4166, 2606, 1202, 817, 352, 166, 81, 36, 15, 10016, 10416

Offset: 1

Clark Kimberling, Jun 13 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....6....16....41
3....7....17...42....106
4....11...27...67....167
5....12...31...77....192
8....21...52...131...327
9....22...56...141...352

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047205, A047216, A191736, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047216 *)
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}]] (* A191726 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191726  *)

A191727 Dispersion of A047219, (numbers >1 and congruent to 1 or 3 mod 5), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 8, 6, 4, 21, 16, 11, 5, 53, 41, 28, 13, 7, 133, 103, 71, 33, 18, 9, 333, 258, 178, 83, 46, 23, 10, 833, 646, 446, 208, 116, 58, 26, 12, 2083, 1616, 1116, 521, 291, 146, 66, 31, 14, 5208, 4041, 2791, 1303, 728, 366, 166, 78, 36, 15, 13021, 10103

Offset: 1

Clark Kimberling, Jun 13 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....3....8....21....53
2....6....16...41....103
4....11...28...71....178
5....13...33...83....208
7....18...46...116...291
9....23...58...146...366

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

Cf. A047212, A047219, A191737, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047219 *)
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}]] (* A191727 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191727  *)

A191728 Dispersion of A047209, (numbers >1 and congruent to 1 or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 11, 6, 3, 29, 16, 9, 5, 74, 41, 24, 14, 7, 186, 104, 61, 36, 19, 8, 466, 261, 154, 91, 49, 21, 10, 1166, 654, 386, 229, 124, 54, 26, 12, 2916, 1636, 966, 574, 311, 136, 66, 31, 13, 7291, 4091, 2416, 1436, 779, 341, 166, 79, 34, 15, 18229, 10229

Offset: 1

Clark Kimberling, Jun 13 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....4....11...29....74
2....6....16...41....104
3....9....24...61....154
5....14...36...91....229
7....19...49...124...311
8....21...54...136...341

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A047222, A047209, A191738, A191722, A191426.

(* Program generates the dispersion array t of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=4; b=6; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047209 *)
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}]] (* A191728 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191728  *)

A191729 Dispersion of A047221, (numbers >1 and congruent to 2 or 3 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 7, 18, 12, 6, 17, 43, 28, 13, 9, 42, 107, 68, 32, 22, 10, 103, 267, 168, 78, 53, 23, 11, 257, 667, 418, 193, 132, 57, 27, 14, 642, 1667, 1043, 482, 328, 142, 67, 33, 15, 1603, 4167, 2607, 1203, 818, 353, 167, 82, 37, 16, 4007, 10417, 6517

Offset: 1

Clark Kimberling, Jun 13 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....3....7.....17
4....8....18...43....107
5....12...28...68....168
6....13...32...78....193
9....22...53...132...328
10...23...57...142...353

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A008854, A047221, A191739, A191722, 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; m[n_]:=If[Mod[n,2]==0,1,0];
f[n_]:=a*m[n+1]+b*m[n]+5*Floor[(n-1)/2]
Table[f[n], {n, 1, 30}]  (* A047221 *)
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}]] (* A191729 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191729  *)

cp's OEIS Frontend

A191722 Dispersion of A008851, (numbers >1 and congruent to 0 or 1 mod 5), by antidiagonals.

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A008548 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+1).

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A003226 Automorphic numbers: m^2 ends with m.

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A191723 Dispersion of A047215, (numbers >1 and congruent to 0 or 2 mod 5), by antidiagonals.

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A191724 Dispersion of A047218, (numbers >1 and congruent to 0 or 3 mod 5), by antidiagonals.

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A191725 Dispersion of A047208, (numbers >1 and congruent to 0 or 4 mod 5), by antidiagonals.

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A191726 Dispersion of A047216, (numbers >1 and congruent to 1 or 2 mod 5), by antidiagonals.

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