keyword:hard - cp's OEIS frontend

A181162 Number of commuting functions: the number of ordered pairs (f,g) of functions from {1..n} to itself such that fg=gf (i.e., f(g(i))=g(f(i)) for all i).

1, 1, 10, 141, 2824, 71565, 2244096, 83982199, 3681265792, 186047433225, 10716241342240, 697053065658411, 50827694884298784, 4129325095108122637, 371782656333674104624, 36918345387693628911375, 4025196918605160943576576, 479796375191949916361466897

Offset: 0

Jeffrey Norden, Oct 07 2010

Also, the total number of endomorphisms of all directed graphs on n labeled vertices with outdegree of each vertex equal 1. - Max Alekseyev, Jan 09 2015
Seems to be relatively hard to compute for large n. (a(n)-n^n)/2 is always an integer, since it gives the number of unordered pairs of distinct commuting functions.
a(n) is divisible by n as proved by Holloway and Shattuck (2015).
From Joerg Arndt, Jul 21 2014: (Start)
Multiply fg=gf from the right by f to obtain fgf=gff, and use f(gf)=f(fg)=ffg to see ffg=gff; iterate to see f^k g = g f^k for all k>=1; by symmetry g^k f = f g^k holds as well.
More generally, if X and Y are words of length w over the alphabet {f,g}, then X = Y (as functional composition) whenever both words contain j symbols f and k symbols g (and j+k=w).  (End)
Functions with the same mapping pattern have the same number of commuting functions, so there is no need to check every pair. - Martin Fuller, Feb 01 2015

			The a(2) = 10 pairs of maps [2] -> [2] are:
01:  [ 1 1 ]  [ 1 1 ]
02:  [ 1 1 ]  [ 1 2 ]
03:  [ 1 2 ]  [ 1 1 ]
04:  [ 1 2 ]  [ 1 2 ]
05:  [ 1 2 ]  [ 2 1 ]
06:  [ 1 2 ]  [ 2 2 ]
07:  [ 2 1 ]  [ 1 2 ]
08:  [ 2 1 ]  [ 2 1 ]
09:  [ 2 2 ]  [ 1 2 ]
10:  [ 2 2 ]  [ 2 2 ]
- _Joerg Arndt_, Jul 22 2014

Martin Fuller, Table of n, a(n) for n = 0..20
Joerg Arndt, the a(3) = 141 pairs of maps [3] -> [3]
Stephen M. Buckley, Minimal order semigroups with specified commuting probability, 04-03-2013. - _W. Edwin Clark_, Jul 21 2014
Martin Fuller, a(6) from the A001372(6)=130 mapping patterns
M. Holloway and M. Shattuck, Commuting pairs of functions on a finite set, 2015.
Math Overflow, What is the probability two random maps on n symbols commute?, 2013. - _W. Edwin Clark_, Jul 21 2014
Math Overflow, Counting and understanding commuting functions, 2010.

A053529 is a similar count for permutations. A254529 is for permutations commuting with functions.

Cf. A000312, A001372, A239749-A239785, A239836-A239841.

(* This brute force code allows to get a few terms *)
a[n_] := a[n] = If[n == 0, 1, Module[{f, g, T}, T = Tuples[Range[n], n]; Table[f = T[[j, #]]&; g = T[[k, #]] &; Table[True, {n}] == Table[f[g[i]] == g[f[i]], {i, n}], {j, n^n}, {k, n^n}] // Flatten // Count[#, True]&]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 5}] (* Jean-François Alcover, Sep 24 2022 *)

a(11)-a(20) from Martin Fuller, Feb 01 2015

A002234 Numbers k such that the Woodall number k*2^k - 1 is prime.

Original entry on oeis.org

2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, 15822, 18885, 22971, 23005, 98726, 143018, 151023, 667071, 1195203, 1268979, 1467763, 2013992, 2367906, 3752948, 17016602

Offset: 1

N. J. A. Sloane, Simon Plouffe

a(34) = 17016602 is tentative until the range 16838832..17016601 is fully searched. - Eric W. Weisstein, Mar 22 2018

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 115, p. 40, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B20.
F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 95, 1983.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 241-242.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 139.

Ray Ballinger, Woodall Primes: Definition and Status.
Ray Ballinger and Wilfrid Keller, Woodall numbers.
C. K. Caldwell, Woodall Numbers.
J. DeMaio, Generalized Woodall Numbers.
Brady Haran and Matt Parker, 383 is cool, Numberphile video (2017).
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
PrimeGrid, PrimeGrid Primes: Subproject: (WOO) Woodall Prime Search.
Matt Parker and Brady Haran, 383 and Woodall Primes, Numberphile video (2017).
H. Riesel, Lucasian criteria for the primality of N=h.2^n-1, Math. Comp., 23 (1969), 869-875.
Eric Weisstein's World of Mathematics, Woodall Number.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Cf. A050918 (for the actual primes), A003261, A005849.

is(n)=isprime(n<Charles R Greathouse IV, Feb 07 2017

a(27) communicated by Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 15 2004
a(28) = 1195203 found by M. Rodenkirch; contributed by Eric W. Weisstein, Nov 29 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(30)-a(33) from John Blazek, May 14 2009
a(34) = 17016602 communicated by Eric W. Weisstein, Mar 22 2018

A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

Original entry on oeis.org

2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 249037, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111

Offset: 1

N. J. A. Sloane

In other words, smallest prime which is at the end of an arithmetic progression of n primes.
For the corresponding values of the first term and the common difference, see A113827 and A093364. For the actual arithmetic progressions, see A133277.
One may also minimize the common difference: this leads to A033189, A033188 and A113872.
One may also specify that the first term is the n-th prime and then minimize the common difference (or, equally, the last term): this leads to A088430 and A113834.
One may also ask for n consecutive primes in arithmetic progression: this gives A006560.

			n, AP, last term
1 2 2
2 2+j 3
3 3+2j 7
4 5+6j 23
5 5+6j 29
6 7+30j 157
7 7+150j 907
8 199+210j 1669
9 199+210j 1879
10 199+210j 2089
11 110437+13860j 249037
12 110437+13860j 262897
..........................
a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.

R. K. Guy, Unsolved Problems in Number Theory, A5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
Thomas Bloom, Problem 200 and Problem 215, Erdős Problems.
Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.
Ben Green and Terence Tao, A bound for progressions of length k in the primes
Andrew Granville, Prime number patterns, American Mathematical Monthly 115 (2008), pp. 279-296.
Paul A. Pritchard, Andrew Moran and Anthony Thyssen, Twenty-two primes in arithmetic progression, Math. Comp. 64 (1995), no. 211, 1337-1339.
Terence Tao, Erdős problem database, see no. 200, 215.
Index entries for sequences related to primes in arithmetic progressions

For the associated gaps, see A093364. For the initial terms, see A113827. For the arithmetic progressions, see A133277.

Cf. A006560, A096003, A113830-A113834, A088430.

(* This program will generate the 4 to 12 terms to use a[n_] to generate term 13 or higher, it will have a prolonged run time. *) a[n_] := Module[{i, p, found, j, df, k}, i = 1; While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > ((n - 1)*df)) && (found == 0), found = 1; Do[If[! PrimeQ[p - k*df], found = 0], {k, 1, n - 1}]]; found == 0]; p]; Table[a[i], {i, 4, 12}]

Green & Tao prove that this sequence is infinite, and further a(n) < 2^2^2^2^2^2^2^2^O(n). Granville conjectures that a(n) <= n! + 1 for n >= 3 and give a heuristic suggesting a(n) is around (exp(1-gamma) n/2)^(n/2). - Charles R Greathouse IV, Feb 26 2013

a(11)-a(13) from Michael Somos, Mar 14 2004
a(14) and corrected version of a(7) from Hugo Pfoertner, Apr 27 2004
a(15)-a(17) from Don Reble, Apr 27 2004
a(18)-a(21) from Granville's paper, Jan 26 2006
Entry revised by N. J. A. Sloane, Jan 26 2006, Oct 17 2007

A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 6, 16, 59, 265, 1544, 10778, 88168, 805491, 8037418, 86221634, 985870522, 11946487647, 152808063181, 2056692014474, 29051272833609, 429668180677439, 6640165204855036, 107026584471569605, 1796101588825595008, 31333997930603283531, 567437240683788292989

Offset: 0

N. J. A. Sloane

The null graph on 0 vertices is vacuously connected and 4-regular. - Jason Kimberley, Jan 29 2011
The Multiset Transform of this sequence gives a triangle which gives in row n and column k the 4-regular simple graphs with n>=1 nodes and k>=1 components (row sums A033301), starting:
;
;
;
;
1 ;
1 ;
2 ;
6 ;
16 ;
59 1 ;
265 1 ;
1544 3 ;
10778 8 ;
88168 25 ;
805491 87 1 ;
8037418 377 1 ;
86221634 2023 3 ;
985870522 13342 9 ;
11946487647 104568 27 ;
152808063181 930489 96 1 ;  - R. J. Mathar, Jun 02 2022

CRC Handbook of Combinatorial Designs, 1996, p. 648.
I. A. Faradzev, Constructive enumeration of combinatorial objects, pp. 131-135 of Problèmes combinatoires et théorie des graphes (Orsay, 9-13 Juillet 1976). Colloq. Internat. du C.N.R.S., No. 260, Centre Nat. Recherche Scient., Paris, 1978.
R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford, 1998.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Wayne Barrett, Shaun Fallat, Veronika Furst, Shahla Nasserasr, Brendan Rooney, and Michael Tait, Regular Graphs of Degree at most Four that Allow Two Distinct Eigenvalues, arXiv:2305.10562 [math.CO], 2023. See p. 7.
Jason Kimberley, Index of sequences counting connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
M. Meringer, GenReg, Generation of regular graphs, program.
Markus Meringer, H. James Cleaves, Stephen J. Freeland, Beyond Terrestrial Biology: Charting the Chemical Universe of α-Amino Acid Structures, Journal of Chemical Information and Modeling, 53.11 (2013), pp. 2851-2862.
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Quartic Graph
Eric Weisstein's World of Mathematics, Regular Graph
Zhipeng Xu, Xiaolong Huang, Fabian Jimenez, and Yuefan Deng, A new record of enumeration of regular graphs by parallel processing, arXiv:1907.12455 [cs.DM], 2019.

From Jason Kimberley, Mar 27 2010 and Jan 29 2011: (Start)
4-regular simple graphs: this sequence (connected), A033483 (disconnected), A033301 (not necessarily connected).
Connected regular simple graphs: A005177 (any degree), A068934 (triangular array); specified degree k: A002851 (k=3), this sequence (k=4), A006821 (k=5), A006822 (k=6), A014377 (k=7), A014378 (k=8), A014381 (k=9), A014382 (k=10), A014384 (k=11).
Connected 4-regular simple graphs with girth at least g: this sequence (g=3), A033886 (g=4), A058343 (g=5), A058348 (g=6).
Connected 4-regular simple graphs with girth exactly g: A184943 (g=3), A184944 (g=4), A184945 (g=5).
Connected 4-regular graphs: this sequence (simple), A085549 (multigraphs with loops allowed), A129417  (multigraphs with loops verboten). (End)

a(n) = A184943(n) + A033886(n).
a(n) = A033301(n) - A033483(n).
Inverse Euler transform of A033301.
Row sums of A184940. - R. J. Mathar, May 30 2022

a(19)-a(22) were appended by Jason Kimberley on Sep 04 2009, Nov 24 2009, Mar 27 2010, and Mar 18 2011, from running M. Meringer's GENREG for 3.4, 44, and 403 processor days, and 15.5 processor years, at U. Ncle.

a(22) corrected and a(23)-a(28) from Andrew Howroyd, Mar 10 2020

A065199 Record high values in A033665, ignoring those numbers that are believed never to reach a palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 24, 30, 53, 54, 55, 58, 64, 78, 79, 80, 82, 96, 97, 98, 109, 112, 113, 131, 135, 147, 149, 186, 187, 188, 198, 201, 232, 233, 236, 259, 260, 261

Offset: 1

Klaus Brockhaus, Oct 20 2001

Records for the number of 'Reverse and Add' steps needed to reach a palindrome.

A065198 gives the corresponding starting points.

			Starting with 89, 24 'Reverse and Add' steps are needed to reach a palindrome; starting with n < 89, at most 6 steps are needed.
For n = A065198(21) = 1005499526, a(21) = 109 "reverse and add" operations are needed to reach a palindrome; for all smaller n, at most 98 steps are needed.
For n = A065198(31) ~ 10^14, a(31) = 198 "reverse and add" operations are needed to reach a palindrome; for all smaller n, at most 188 steps are needed.
For n = A065198(36) ~ 10^18, a(36) = 259 "reverse and add" operations are needed to reach a palindrome; for all smaller n, at most 236 steps are needed.

Jason Doucette, World records
Index entries for sequences related to Reverse and Add!

Cf. A033665, A033865, A023109, A065198.

limit = 10^3; (* Assumes that there is no palindrome if none is found before "limit" iterations *)
best = -1; lst = {};
For[n = 0, n <= 100000, n++,
np = n; i = 0;
While[np != IntegerReverse[np] && i < limit,
  np = np + IntegerReverse[np]; i++];
If[i < limit && i > best, best = i; AppendTo[lst, i]]]; lst (* Robert Price, Oct 14 2019 *)

my(m, M=-1); for(n=0, oo, (MA033665(n, M+39))&&print1(M=m", ")) \\ For illustration; becomes very slow for terms > 70, even with the "custom" search limit as optional 2nd arg to A033665. - M. F. Hasler, Feb 16 2020

a(n) = A033665(A065198(n)). - M. F. Hasler, Feb 16 2020

Terms a(17) to a(21) from Sascha Kurz, Dec 05 2001
Terms a(22) onwards were taken from Jason Doucette, World records. - Klaus Brockhaus, Sep 24 2003
Terms a(36) to a(38) were taken from Jason Doucette, World records and added by A.H.M. Smeets, Feb 10 2019
Edited by N. J. A. Sloane, Jul 16 2021

A128027 Numbers n such that (11^n - 3^n)/8 is prime.

Original entry on oeis.org

3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397

Offset: 1

Alexander Adamchuk, Feb 11 2007

All terms are primes.

No other terms < 10^5.

Cf. A028491 = numbers n such that (3^n - 1)/2 is prime. Cf. A057468 = numbers n such that 3^n - 2^n is prime. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128028, A128029, A128030, A128031, A128032.

[p: p in PrimesUpTo(400) | IsPrime((11^p-3^p) div 8)]; // Vincenzo Librandi, Nov 20 2014

A128027:=n->`if`(isprime((11^n-3^n)/8),n,NULL): seq(A128027(n),n=1..1000); # Wesley Ivan Hurt, Nov 19 2014

k=8; Select[ Prime[ Range[1,200] ], PrimeQ[ ((k+3)^# - 3^#)/k ]& ]
Do[If[PrimeQ[(11^n - 3^n)/8], Print[n]], {n, 10^4}] (* Ryan Propper, Mar 17 2007 *)
Select[Prime[Range[1200]], PrimeQ[(11^# - 3^#)/8] &] (* Vincenzo Librandi, Nov 20 2014 *)

is(n)=ispseudoprime((11^n - 3^n)/8) \\ Charles R Greathouse IV, Feb 17 2017

a(8) from Ryan Propper, Mar 17 2007
a(9) from Farideh Firoozbakht, Apr 04 2007
a(10)=13103, a(11)=90397 from Robert Price, Apr 24 2011

A128154 a(n) = least k such that the remainder when 14^k is divided by k is n.

Original entry on oeis.org

13, 3, 11, 5, 33, 10, 1967, 9, 23587, 18, 2733, 46, 17651, 15, 93929, 20, 303, 178, 145, 22, 12901, 58, 2721, 25, 17990951, 27, 143, 36, 85, 166, 646123, 82, 2439143677, 55, 63, 76, 319, 123, 295, 52, 51, 77, 247380287953, 45, 5779134947, 90, 87, 74, 175, 146

Offset: 1

Alexander Adamchuk, Feb 16 2007

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127817, A127818, A127819, A127820, A127821, A128155, A128156, A128157, A128158, A128159, A128160, A128149, A128150.

t = Table[0, {10000} ]; k = 1; While[ k < 3000000000, a = PowerMod[14, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t
lk[n_]:=Module[{k=1},While[PowerMod[14,k,k]!=n,k++];k]; Array[lk,20] (* Harvey P. Dale, Aug 17 2013 *)

More terms from Ryan Propper, Feb 28 2007

a(43) from Hagen von Eitzen, Aug 16 2009

A001930 Number of topologies, or transitive digraphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 3, 9, 33, 139, 718, 4535, 35979, 363083, 4717687, 79501654, 1744252509, 49872339897, 1856792610995, 89847422244493, 5637294117525695

Offset: 0

N. J. A. Sloane

			From _Gus Wiseman_, Aug 02 2019: (Start)
Non-isomorphic representatives of the a(0) = 1 through a(3) = 9 topologies:
  {}  {}{1}  {}{12}        {}{123}
             {}{2}{12}     {}{3}{123}
             {}{1}{2}{12}  {}{23}{123}
                           {}{1}{23}{123}
                           {}{3}{23}{123}
                           {}{2}{3}{23}{123}
                           {}{3}{13}{23}{123}
                           {}{2}{3}{13}{23}{123}
                           {}{1}{2}{3}{12}{13}{23}{123}
(End)

Loic Foissy, Claudia Malvenuto, Frederic Patras, Infinitesimal and B_infinity-algebras, finite spaces, and quasi-symmetric functions, Journal of Pure and Applied Algebra, Elsevier, 2016, 220 (6), pp. 2434-2458. .
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 218 (but the last entry is wrong).
M. Kolli, On the cardinality of the T_0-topologies on a finite set, Preprint, 2014.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. A. Wright, There are 718 6-point topologies, quasi-orderings and transgraphs, Notices Amer. Math. Soc., 17 (1970), p. 646, Abstract #70T-A106.
J. A. Wright, personal communication.
For further references concerning the enumeration of topologies and posets see under A000112 and A001035.

C. M. Bender et al., Combinatorics and field theory, arXiv:quant-ph/0604164, 2006.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5
Gunnar Brinkmann and Brendan D. McKay, Counting unlabeled topologies and transitive relations.
G. Brinkmann and B. D. McKay, Counting unlabeled topologies and transitive relations, J. Integer Sequences, Volume 8, 2005.
Gunnar Brinkmann and Brendan D. McKay, Counting Unlabelled Topologies and Transitive Relations, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.1.
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184
K. K.-H. Butler and G. Markowsky, Enumeration of finite topologies, Proc. 4th S-E Conf. Combin., Graph Theory, Computing, Congress. Numer. 8 (1973), 169-184. [Annotated scan of pages 180 and 183 only]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. R. Finch, Transitive relations, topologies and partial orders
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
L. Foissy, C. Malvenuto, and F. Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014-2015.
Misha Gavrilovich and Misha Rabinovich, The Quillen negation monoid of a category, and Schreier graphs of its action on classes of morphisms, 2024. See p. 11.
Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550, 2012. - From _N. J. A. Sloane_, Nov 09 2012
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
G. Pfeiffer, Counting Transitive Relations, preprint, 2004.
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
D. Rusin, Further information and references [Broken link]
D. Rusin, Further information and references [Cached copy]
Henry Sharp, Jr., Quasi-orderings and topologies on finite sets, Proceedings of the American Mathematical Society 17.6 (1966): 1344-1349. [Annotated scanned copy]
N. J. A. Sloane, List of sequences related to partial orders, circa 1972
N. J. A. Sloane, Classic Sequences
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 8 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Swartz and Nicholas J. Werner, Zero pattern matrix rings, reachable pairs in digraphs, and Sharp's topological invariant tau, arXiv:1709.05390 [math.CO], 2017.
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Digraph Topology.
R. H. Warren, The number of topologies, Houston J. Math., 8 (No. 2, 1982), 297-301. Mentions a(4)=33. [Annotated scanned copy]
Wikipedia Topological space
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences

Cf. A000798 (labeled topologies), A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057, A001928, A001929.
The case with unions only is A108798.
The case with intersections only is (also) A108798.
Partial sums are A326898 (the non-covering case).
Cf. A000612, A003180, A108800, A193674, A306445, A326876, A326878, A326882.

a(8)-a(12) from Goetz Pfeiffer (goetz.pfeiffer(AT)nuigalway.ie), Jan 21 2004

a(13)-a(16) from Brinkmann's and McKay's paper, sent by Vladeta Jovovic, Jan 04 2006

A005315 Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.

Original entry on oeis.org

1, 1, 2, 8, 42, 262, 1828, 13820, 110954, 933458, 8152860, 73424650, 678390116, 6405031050, 61606881612, 602188541928, 5969806669034, 59923200729046, 608188709574124, 6234277838531806, 64477712119584604, 672265814872772972, 7060941974458061392

Offset: 0

N. J. A. Sloane, J. A. Reeds (reeds(AT)idaccr.org)

There is a 1-to-1 correspondence between loops crossing a road 2n times and lines crossing a road 2n-1 times.

S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303.
S. K. Lando and A. K. Zvonkin, Meanders, Selecta Mathematica Sovietica, Vol. 11, Number 2, pp. 117-144, 1992.
A. Phillips, Simple Alternating Transit Mazes, preprint. Abridged version appeared as "La topologia dei labirinti," in M. Emmer, editor, L'Occhio di Horus: Itinerari nell'Imaginario Matematico. Istituto della Enciclopedia Italia, Rome, 1989, pp. 57-67.
V. R. Pratt, personal communication.
J. A. Reeds and L. A. Shepp, An upper bound on the meander constant, preprint, May 25, 1999. [Obtains upper bound of 13.01]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
For additional references, see A005316.

Andrew Howroyd, Table of n, a(n) for n = 0..28 (first 24 terms from Iwan Jensen)
Oswin Aichholzer, Carlos Alegría Galicia, Irene Parada, Alexander Pilz, Javier Tejel, Csaba D. Tóth, Jorge Urrutia, and Birgit Vogtenhuber, Hamiltonian meander paths and cycles on bichromatic point sets, XVIII Spanish Meeting on Computational Geometry (Girona, 2019).
V. I. Arnol'd, A branched covering of CP^2->S^4, hyperbolicity and projective topology [ Russian ], Sibir. Mat. Zhurn., 29 (No. 2, 1988), 36-47 = Siberian Math. J., 29 (1988), 717-725.
Roland Bacher, Meander algebras
David Bevan, Random Closed Meanders - _David Bevan_, Jun 25 2010
Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, Sampling planar tanglegrams and pairs of disjoint triangulations, arXiv:2304.05318 [math.CO], 2023.
B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, Transactions on Algorithms, Vol. 6 No. 2 (2010) article #42, 12 pages.
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 525.
Reinhard O. W. Franz, and Berton A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), no. 1, 7-17.
Erich Friedman, Illustration of initial terms
Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
Iwan Jensen, Home page
Iwan Jensen, Closed meanders, a(n) for n = 1..24
Iwan Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
Iwan Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33, 5953-5963 (2000).
Iwan Jensen and A. J. Guttmann, Critical exponents of plane meanders J. Phys. A 33, L187-L192 (2000).
Michael La Croix, Approaches to the Enumerative Theory of Meanders, 2003.
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
A. Panayotopoulos, On Meandric Colliers, Mathematics in Computer Science, (2018).
A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.
A. Phillips, Mazes
A. Phillips, Simple, Alternating, Transit Mazes
J. A. Reeds, D. E. Knuth, & N. J. A. Sloane, Email Correspondence
J. Reeds, L. Shepp, & D. McIlroy, Numerical bounds for the Arnol'd "meander" constant, Preprint.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

These are the odd-numbered terms of A005316. Cf. A077054. For nonisomorphic solutions, see A077460.

A column of triangle A008828.

A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 0, 1, A005316[[2n]]];
a /@ Range[0, 28] (* Jean-François Alcover, Sep 25 2019 *)

a(n) = A005316(2n-1) for n>0.

A180414 Number of different resistances that can be obtained by combining n one-ohm resistors.

Original entry on oeis.org

1, 2, 4, 8, 16, 36, 80, 194, 506, 1400, 4039, 12044, 36406, 111324, 342447, 1064835, 3341434, 10583931, 33728050, 107931849, 346616201

Offset: 0

Vaclav Kotesovec, Sep 02 2010

In "addendum" J. Karnofsky stated the value a(15) = 1064833. In contrast to the terms up to and including a(14), which could all be confirmed, an independent calculation based on a list of 3-connected simple graphs resulted in the corrected value a(15) = 1064835. - Hugo Pfoertner, Dec 06 2020

See A337517 for the number of different resistances that can be obtained by combining /exactly/ n one-ohm resistors. The method used by Andrew Howroyd (see his program in the link section) uses 3-connected graphs with one edge (the 'battery edge') removed. - Rainer Rosenthal, Feb 07 2021

			a(n) counts all resistances that can be obtained with fewer than n resistors as well as with exactly n resistors. Without a resistor the resistance is infinite, i.e., a(0) = 1. One 1-ohm resistor adds resistance 1, so a(1) = 2. Two resistors in parallel give 1/2 ohm, while in series they give 2 ohms. So a(2) is the number of elements in the set {infinity, 1, 1/2, 2}, i.e., a(2) = 4. - _Rainer Rosenthal_, Feb 07 2021

Technology Review's Puzzle Corner, How many different resistances can be obtained by combining 10 one ohm resistors? Oct 3, 2003.

Allan Gottlieb, Oct 3, 2003 addendum (Karnofsky).
Andrew Howroyd, PARI program (includes scripts for other related sequences)
Joel Karnofsky, Solution of problem from Technology Review's Puzzle Corner Oct 3, 2003, Feb 23 2004.
Joel Karnofsky, Mathematica program for resistances, copied from article.
Index to sequences related to resistances.

Cf. A048211, A051389, A174284, A337517, A338197, A338487, A338573, A338580, A338590, A338600.

Cf. A123545, A338511, A338999, A339072.

Mathematica
```
(* See link. *)
```

a(n) = A174284(n) + 1 for n <= 7, a(n) > A174284(n) + 1 otherwise. - Hugo Pfoertner, Nov 01 2020

a(n) is the number of elements in the union of the sets SetA337517(k), k <= n, counted by A337517. - Rainer Rosenthal, Feb 07 2021

a(15) corrected and a(16) added by Hugo Pfoertner, Dec 06 2020
a(17) from Hugo Pfoertner, Dec 09 2020
a(0) from Rainer Rosenthal, Feb 07 2021
a(18) from Hugo Pfoertner, Apr 09 2021
a(19) from Zhao Hui Du, May 15 2023
a(20) from Zhao Hui Du, May 23 2023

cp's OEIS Frontend

A181162 Number of commuting functions: the number of ordered pairs (f,g) of functions from {1..n} to itself such that fg=gf (i.e., f(g(i))=g(f(i)) for all i).

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A002234 Numbers k such that the Woodall number k*2^k - 1 is prime.

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A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

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A006820 Number of connected regular simple graphs of degree 4 (or quartic graphs) with n nodes.

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A065199 Record high values in A033665, ignoring those numbers that are believed never to reach a palindrome.

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A128027 Numbers n such that (11^n - 3^n)/8 is prime.

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A128154 a(n) = least k such that the remainder when 14^k is divided by k is n.

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