keyword:hard - cp's OEIS frontend

A102896 Number of ACI algebras (or semilattices) on n generators with no annihilator.

1, 2, 7, 61, 2480, 1385552, 75973751474, 14087648235707352472

Offset: 0

Mitch Harris, Jan 18 2005

Or, number of Moore families on an n-set, that is, families of subsets that contain the universal set {1,...,n} and are closed under intersection.
Or, number of closure operators on a set of n elements.
An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation.
Also the number of set-systems on n vertices that are closed under union. The BII-numbers of these set-systems are given by A326875. - Gus Wiseman, Jul 31 2019
From Bernhard Ganter, Jul 08 2025: (Start)
Also the number of union-free families of subsets of an n-set; i.e., families of nonempty sets on n elements such that no set is a union of some others.
Also the number of intersection-free families of subsets of an n-set; i.e., of families of proper subsets on n elements such that no set is an intersection of some others.
(Note that every union-free family on an n-set is the set of union-irreducible elements of exactly one union-closed family, and each family of union-irreducible elements is union-free. Same for intersection.) (End)

			From _Gus Wiseman_, Jul 31 2019: (Start)
The a(0) = 1 through a(2) = 7 set-systems closed under union:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}
(End)

G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967.
Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.
P. Colomb, A. Irlande and O. Raynaud, Counting of Moore Families for n=7, International Conference on Formal Concept Analysis (2010). [From Pierre Colomb (pierre(AT)colomb.me), Sep 04 2010]
E. H. Moore, Introduction to a Form of General Analysis, AMS Colloquium Publication 2 (1910), pp. 53-80.

Andrew J. Blumberg, Michael A. Hill, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim, Homotopical Combinatorics, Notices Amer. Math. Soc. (2024) Vol. 71, No. 2, 260-266. See p. 261.
Daniel Borchmann and Bernhard Ganter, Concept Lattice Orbifolds - First Steps, Proceedings of the 7th International Conference on Formal Concept Analysis (ICFCA 2009), 22-37.
Jishnu Bose, Tien Chih, Hannah Housden, Legrand Jones II, Chloe Lewis, Kyle Ormsby, and Millie Rose, Combinatorics of factorization systems on lattices, arXiv:2503.22883 [math.CO], 2025. See p. 11.
Pierre Colomb, Alexis Irlande, Olivier Raynaud and Yoan Renaud, About the Recursive Decomposition of the lattice of co-Moore Families, ICFCA 2011.
Pierre Colomb, Alexis Irlande, Olivier Raynaud, and Yoan Renaud, Recursive decomposition tree of a Moore co-family and closure algorithm, Annals of Mathematics and Artificial Intelligence, 2013, DOI 10.1007/s10472-013-9362-x.
Nachum Dershowitz, Mitchell A. Harris, and Guan-Shieng Huang, Enumeration Problems Related to Ground Horn Theories, arXiv:cs/0610054v2 [cs.LO], 2006-2008.
Michel Habib and Lhouari Nourine, The number of Moore families on n = 6, Discrete Math., 294 (2005), 291-296.
Caoilainn Kirkpatrick, Amelie el Mahmoud, Kyle Ormsby, Angélica M. Osorno, Dale Schandelmeier-Lynch, Riley Shahar, Lixing Yi, Avery Young, and Saron Zhu, Enumerating submonoids of finite commutative monoids, arXiv:2508.20786 [math.CO], 2025. See p. 16.

For set-systems closed under union:
- The covering case is A102894.
- The unlabeled case is A193674.
- The case also closed under intersection is A306445.
- Set-systems closed under overlapping union are A326866.
- The BII-numbers of these set-systems are given by A326875.
Cf. A102895, A102897, A108798, A108800, A193675, A000798, A014466, A326878, A326880, A326881.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],SubsetQ[#,Union@@@Tuples[#,2]]&]],{n,0,3}] (* Gus Wiseman, Jul 31 2019 *)

a(n) = Sum_{k=0..n} C(n, k)*A102894(k), where C(n, k) is the binomial coefficient.
For asymptotics see A102897.
a(n) = A102897(n)/2. - Gus Wiseman, Jul 31 2019

N. J. A. Sloane added a(6) from the Habib et al. reference, May 26 2005
Additional comments from Don Knuth, Jul 01 2005
a(7) from Pierre Colomb (pierre(AT)colomb.me), Sep 04 2010

A175607 Largest number k such that the greatest prime factor of k^2 - 1 is prime(n).

Original entry on oeis.org

3, 17, 161, 8749, 19601, 246401, 672281, 23718421, 10285001, 354365441, 3222617399, 9447152318, 127855050751, 842277599279, 2218993446251, 2907159732049, 41257182408961, 63774701665793, 25640240468751, 238178082107393, 4573663454608289, 19182937474703818751, 34903240221563713, 332110803172167361, 99913980938200001

Offset: 1

Charles R Greathouse IV, Jul 23 2010

For any prime p, there are finitely many k such that k^2-1 has p as its largest prime factor.
For every prime p, is there some k where the greatest prime factor of k^2-1 is p? Answer from Artur Jasinski, Oct 22 2010: Yes.
As mentioned by Luca and Najman, this problem is closely related to the one in A002071.
The terms give an upper bound with a method for the simultaneous computation of logarithms of small primes, see the fxtbook link. - Joerg Arndt, Jul 03 2012

Charles R Greathouse IV, Table of n, a(n) for n = 1..25
Joerg Arndt, Matters Computational (The Fxtbook), section 32.4, pp.632-633.
Florian Luca and Filip Najman, "On the largest prime factor of x^2-1", Mathematics of Computation 80:273 (2011), pp. 429-435. (Paper has errata that was posted on the MOC website.)
Filip Najman, Home Page (gives all 16223 numbers k such that k^2-1 has no prime factor greater than 97)

Cf. A214093 (largest primes p such that the greatest prime factor of p^2-1 is prime(n)).
Cf. A076605 (largest prime divisor of n^2-1).
Cf. A285283 (equivalent for k^2+1). - Tomohiro Yamada, Apr 22 2017
Cf. A006530, A005563. - M. F. Hasler, Jun 13 2018

/* up to term for p=97 */
/* S[] is the list computed by Filip Najman (16223 elements) */
S=[2,3,4, ... ,332110803172167361, 19182937474703818751];
lpf(n)={ vecmax(factor(n)[, 1]) } /* largest prime factor */
{ forprime (p=2, 97,
  t = 0;
  for (n=1,#S, if ( lpf(S[n]^2-1)==p, t=n ) );
  print1(S[t],", ");
);}
/* Joerg Arndt, Jul 03 2012 */

More terms (using Filip Najman's list) by Joerg Arndt, Jul 03 2012

A006558 Start of first run of n consecutive integers with same number of divisors.

Original entry on oeis.org

1, 2, 33, 242, 11605, 28374, 171893, 1043710445721, 2197379769820, 2642166652554075

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

The entry 40311 given by Guy and by Wells is incorrect. - Jud McCranie, Jan 20 2002
a(10) <= 2642166652554075, a(11) <= 17707503256664346, a(12) <= 9827470582657267545. - David Wasserman, Feb 22 2008
a(10) > 10^13. - Giovanni Resta, Jul 13 2015
a(12) <= 3842083249515874843. - Hugo van der Sanden, Sep 20 2022
a(13) <= 34169215324203592637988571. - Hugo van der Sanden, Apr 13 2022
a(14) <= 9721439902882994590514319997146. - Hugo van der Sanden, Jun 14 2022
a(15) <= 80215613469168729088982885848674841. - Natalia Makarova, Sep 18 2022
a(16) <= 37981337212463143311694743672867136611416. - Vladimir Letsko, Mar 17 2017
a(17) <= 768369049267672356024049141254832375543516. - Vladimir Letsko, Sep 12 2017
a(18) <= 488900003598703704335810037459507226590256411. - Vladimir Letsko, Jun 03 2022
a(19) <= 5908388043825578351730345292813071711296723319324. - Vladimir Letsko, Apr 09 2022
a(20) <= 17668887847524548413038893976018715843277693308027547. Vladimir Letsko, May 30 2022
Spătaru proves that the longest such run up to N is at most exp(C*sqrt(log N log log N)) for some constant C, hence a(n) >> exp(exp(W((log^2 n)/C))) which is approximately exp(log^2 n/(2 log log n)). - Charles R Greathouse IV, Feb 06 2023

			33 has four divisors (1, 3, 11, and 33), 34 has four divisors (1, 2, 17, and 34), 35 has four divisors (1, 5, 7, and 35).  These are the first three consecutive numbers with the same number of divisors, so a(3)=33.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 33, pp 12, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, section B18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 87.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, pages 147 and 176.

Pentti Haukkanen, Some computational results concerning the divisor functions d(n) and sigma(n), The Mathematics Student, Vol. 62 Nos. 1-4 (1993) pp. 166-168. See p. 167.
Vladimir A. Letsko, Some new results on consecutive equidivisible integers, arXiv:1510.07081 [math.NT], 2015.
Vladimir A. Letsko and Vasilii Dziubenko On consecutive equidivisible integers (in Russian), Boundaries of knowledge, 2 (45) 2016.
Carlos Rivera, Problem 20: k consecutive numbers with the same number of divisors, The Prime Puzzles and Problems Connection.
Carlos Rivera, Problem 61: problem 20 revisited, The Prime Puzzles and Problems Connection.
Vlad-Titus Spătaru, Runs of consecutive integers having the same number of divisors, arXiv preprint (2023). arXiv:2301.04464 [math.NT]

Cf. A000005, A005237, A005238, A006601, A049051, A019273, A039665.

Cf. A034173, A115158, A119479.

tau = DivisorSigma[0, #]&;
A006558[q_, w_] := Module[{a, k, j, ok, n}, For[j = 0, j <= w, j++, For[n = 1, n <= q, n++, ok = 1; a = tau[n]; For[k = 1, k <= j, k++, If[a != tau[n + k], ok = 0; Break[]]]; If [ok == 1, Print[n]; Break[]]]]];
A006558[2*10^5, 7] (* Jean-François Alcover, Dec 10 2017 *)

isok(n, k)=nb = numdiv(k); for (j=k+1, k+n-1, if (numdiv(j) != nb, return(0));); 1;
a(n) = {k=1; while (!isok(n, k), k++); k;} \\ Michel Marcus, Feb 17 2016

a(8) from Jud McCranie, Jan 20 2002
a(9) conjectured by David Wasserman, Jan 08 2006
a(9) confirmed by Jud McCranie, Jan 14 2006
a(10) by Jud McCranie, Nov 27 2018

A127106 Numbers k such that k^2 divides 6^k-1.

Original entry on oeis.org

1, 5, 1555, 9673655, 187159211791705, 776119592182705

Offset: 1

Alexander Adamchuk, Jan 05 2007

Subsequence of A014946.

Cf. A127100, A127101, A127102, A127103, A127104, A127105, A127107, A127092, A014946.

Select[Range[20000], IntegerQ[(PowerMod[6, #, #^2 ]-1)/#^2 ]&]

Two more terms from Max Alekseyev, May 05 2010

A377031 Numbers k such that (27^k - 2^k)/25 is prime.

Original entry on oeis.org

2, 3, 269, 401, 631, 701, 1321, 2707, 5471, 6581

Offset: 1

Robert Price, Oct 13 2024

The definition implies that k must be a prime.

a(11) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161, A375236.

Select[Prime[Range[1000]], PrimeQ[(27^# - 2^#)/25] &]

A002860 Number of Latin squares of order n; or labeled quasigroups.

Original entry on oeis.org

1, 2, 12, 576, 161280, 812851200, 61479419904000, 108776032459082956800, 5524751496156892842531225600, 9982437658213039871725064756920320000, 776966836171770144107444346734230682311065600000

Offset: 1

N. J. A. Sloane

Also the number of minimum vertex colorings in the n X n rook graph. - Eric W. Weisstein, Mar 02 2024

David Nacin, "Puzzles, Parity Maps, and Plenty of Solutions", Chapter 15, The Mathematics of Various Entertaining Subjects: Volume 3 (2019), Jennifer Beineke & Jason Rosenhouse, eds. Princeton University Press, Princeton and Oxford, p. 245.
Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.
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).

Ronald Alter, Research Problems: How Many Latin Squares are There?, Amer. Math. Monthly 82 (1975), no. 6, 632-634. MR1537769.
Stanley E. Bammel and Jerome Rothstein, The number of 9 X 9 Latin squares, Discrete Math., 11 (1975), 93-95.
Daniel Berend, On the number of Sudoku squares, Discrete Mathematics 341.11 (2018): 3241-3248. See p. 3241.
Jeranfer Bermúdez, Richard García, Reynaldo López and Lourdes Morales, Some Properties of Latin Squares, Laboratorio Emmy Noether, 2009.
Matvey Borodin, Eric Chen, Aidan Duncan, Tanya Khovanova, Boyan Litchev, Jiahe Liu, Veronika Moroz, Matthew Qian, Rohith Raghavan, Garima Rastogi, and Michael Voigt, Sequences of the Stable Matching Problem, arXiv:2201.00645 [math.HO], 2021.
John W. Brown, Enumeration of Latin squares with application to order 8, J. Combin. Theory, 5 (1968), 177-184.
Nikhil Byrapuram, Hwiseo (Irene) Choi, Adam Ge, Selena Ge, Tanya Khovanova, Sylvia Zia Lee, Evin Liang, Rajarshi Mandal, Aika Oki, Daniel Wu, and Michael Yang, Quad Squares, arXiv:2308.07455 [math.HO], 2023.
Gloria S. Choi, Beyond Sudoku: New puzzle variants based on a novel mathematical mapping, J. High Sch. Sci. (2025) Vol. 9, No. 2, 70-84.
Hai-Dang Dau and Nicolas Chopin, Waste-free Sequential Monte Carlo, arXiv:2011.02328 [stat.CO], 2020.
Abdelrahman Desoky, Hany Ammar, Gamal Fahmy, Shaker El-Sappagh, Abdeltawab Hendawi, and Sameh H. Basha, Latin Square and Artificial Intelligence Cryptography for Blockchain and Centralized Systems, Int'l Conf. Adv. Intel. Sys. Informat., Proc. 9th Int'l Conf. (AISI 2023) pp. 444-455.
Thangavelu Geetha, Amritanshu Prasad, and Shraddha Srivastava, Schur Algebras for the Alternating Group and Koszul Duality, arXiv:1902.02465 [math.RT], 2019.
E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189-198. MR0179095 (31 #3346). Mentions this sequence. - _N. J. A. Sloane_, Mar 15 2014
Yue Guan, Minjia Shi and Denis S. Krotov, The Steiner triple systems of order 21 with a transversal subdesign TD(3,6), arXiv:1905.09081 [math.CO], 2019.
Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, Fun with Latin Squares, arXiv:2109.01530 [math.HO], 2021.
Yang-Hui He and Minhyong Kim, Learning Algebraic Structures: Preliminary Investigations, arXiv:1905.02263 [cs.LG], 2019.
A.-A. A. Jucys, The number of distinct Latin squares as a group-theoretical constant, J. Combinatorial Theory Ser. A 20 (1976), no. 3, 265-272. MR0419259 (54 #7283).
Dieter Jungnickel and Vladimir D. Tonchev, Counting Steiner triple systems with classical parameters and prescribed rank, arXiv:1709.06044 [math.CO], 2017.
Lintao Liu, Xuehu Yan, Yuliang Lu, and Huaixi Wang, 2-threshold Ideal Secret Sharing Schemes Can Be Uniquely Modeled by Latin Squares, National University of Defense Technology, Hefei, China, (2019).
Brendan D. McKay, Alison Meynert and Wendy Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Des. 15 (2007), no. 2, 98-119.
Brendan D. McKay and Eric Rogoyski, Latin squares of order ten, Electron. J. Combinatorics, 2 (1995) #N3.
Brendan D. McKay and Ian M. Wanless, On the number of Latin squares. Preprint 2004.
Brendan D. McKay and Ian M. Wanless, On the number of Latin squares, Ann. Combinat. 9 (2005) 335-344.
Jia-yu Shao and Wan-di Wei, A formula for the number of Latin squares., Discrete Mathematics 110 (1992) 293-296.
Minjia Shi, Li Xu, and Denis S. Krotov, The number of the non-full-rank Steiner triple systems, arXiv:1806.00009 [math.CO], 2018.
Douglas S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
Douglas S. Stones and Ian M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.
Eric Weisstein's World of Mathematics, Latin Square.
Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
Eric Weisstein's World of Mathematics, Rook Graph.
Mark B. Wells, The number of Latin squares of order 8, J. Combin. Theory, 3 (1967), 98-99.
Krasimir Yordzhev, The bitwise operations in relation to obtaining Latin squares, arXiv preprint arXiv:1605.07171 [cs.OH], 2016.
Index entries for sequences related to Latin squares and rectangles
Index entries for sequences related to quasigroups

Cf. A000315, A000479.
Cf. A003090, A040082, A057991.
Cf. A098679 (Latin cubes).
A row of the array in A249026.

Table[Length[ResourceFunction["FindProperColorings"][GraphProduct[CompleteGraph[n], CompleteGraph[n], "Cartesian"], n]], {n, 5}]

a(n) = n!*A000479(n) = n!*(n-1)!*A000315(n).

One more term (from the McKay-Wanless article) from Richard Bean, Feb 17 2004

A003182 Dedekind numbers: inequivalent monotone Boolean functions of n or fewer variables, or antichains of subsets of an n-set.

Original entry on oeis.org

2, 3, 5, 10, 30, 210, 16353, 490013148, 1392195548889993358, 789204635842035040527740846300252680

Offset: 0

N. J. A. Sloane

NP-equivalence classes of unate Boolean functions of n or fewer variables.
Also the number of simple games with n players in minimal winning form up to isomorphism. - Fabián Riquelme, Mar 13 2018
The labeled case is A000372. - Gus Wiseman, Feb 23 2019
First differs from A348260(n + 1) at a(5) = 210, A348260(6) = 233. - Gus Wiseman, Nov 28 2021
Pawelski & Szepietowski show that a(n) = A001206(n) (mod 2) and that a(9) = 6 (mod 210). - Charles R Greathouse IV, Feb 16 2023

			From _Gus Wiseman_, Feb 20 2019: (Start)
Non-isomorphic representatives of the a(0) = 2 through a(3) = 10 antichains:
  {}    {}     {}         {}
  {{}}  {{}}   {{}}       {{}}
        {{1}}  {{1}}      {{1}}
               {{1,2}}    {{1,2}}
               {{1},{2}}  {{1},{2}}
                          {{1,2,3}}
                          {{1},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}
(End)

I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38.
Arocha, Jorge Luis (1987) "Antichains in ordered sets" [ In Spanish ]. Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico 27: 1-21.
J. Berman, Free spectra of 3-element algebras, in R. S. Freese and O. C. Garcia, editors, Universal Algebra and Lattice Theory (Puebla, 1982), Lect. Notes Math. Vol. 1004, 1983.
G. Birkhoff, Lattice Theory. American Mathematical Society, Colloquium Publications, Vol. 25, 3rd ed., Providence, RI, 1967, p. 63.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 273.
M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 188.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
W. F. Lunnon, The IU function: the size of a free distributive lattice, pp. 173-181 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
Saburo Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38, Table 2.3.2. - Row 13.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. H. Wiedemann, personal communication.

K. S. Brown, Dedekind's problem
Aniruddha Biswas and Palash Sarkar, Counting Unate and Monotone Boolean Functions Under Restrictions of Balancedness and Non-Degeneracy, J. Int. Seq. (2025) Vol. 28, Art. No. 25.3.4. See pp. 2, 14.
Patrick De Causmaecker and Stefan De Wannemacker, On the number of antichains of sets in a finite universe, arXiv:1407.4288 [math.CO], 2014.
Patrick De Causmaecker and Lennart Van Hirtum, Solving systems of equations on antichains for the computation of the ninth Dedekind Number, arXiv:2405.20904 [math.CO], 2024. See p. 4.
Liviu Ilinca and Jeff Kahn, Counting maximal antichains and independent sets, arXiv:1202.4427 [math.CO], 2012; Order 30.2 (2013): 427-435.
J. L. King, Brick tiling and monotone Boolean functions
D. J. Kleitman, On Dedekind's problem: The number of monotone Boolean functions, Proc. Amer. Math. Soc. 21 1969 677-682.
D. J. Kleitman and G. Markowsky, On Dedekind's problem: the number of isotone Boolean functions. II, Trans. Amer. Math. Soc. 213 (1975), 373-390.
Sascha Kurz, Competitive learning of monotone Boolean functions, arXiv:1401.8135 [cs.DS], 2014.
C. L. Mallows, Emails to N. J. A. Sloane, Jun-Jul 1991
Mikaël Monet and Dan Olteanu, Towards Deterministic Decomposable Circuits for Safe Queries, 2018.
S. Muroga, Threshold Logic and Its Applications, Wiley, NY, 1971 [Annotated scans of a few pages]
Bartlomiej Pawelski, On the number of inequivalent monotone Boolean functions of 8 variables, arXiv:2108.13997 [math.CO], 2021. See Table 2 p. 2.
Bartlomiej Pawelski, On the number of inequivalent monotone Boolean functions of 9 variables, arXiv:2305.06346 [math.CO], 2023.
Bartłomiej Pawelski, Counting and generating monotone Boolean functions, Doctoral Diss., Univ. Gdańsk, (Poland, 2024). See pp. 12, 24, 34-35, 40, 49-50.
Bartlomiej Pawelski and Andrzej Szepietowski, Divisibility properties of Dedekind numbers, arXiv:2302.04615 [math.CO], 2023.
Tamon Stephen and Timothy Yusun, Counting inequivalent monotone Boolean functions, Discrete Applied Mathematics, 167 (2014), 15-24.
Tamon Stephen and Timothy Yusun, Counting inequivalent monotone Boolean functions, arXiv preprint arXiv:1209.4623 [cs.DS], 2012.
Andrzej Szepietowski, Fixes of permutations acting on monotone Boolean functions, arXiv:2205.03868 [math.CO], 2022. See p. 17.
Eric Weisstein's World of Mathematics, Boolean Function.
Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.
Index entries for sequences related to Boolean functions

Cf. A000372, A007153, A006602, A007411.
Cf. A006126, A014466, A261005, A293606, A293993, A304996, A305000, A305857, A306505, A319721, A320449, A321679.
Cf. A007363, A046165, A306007, A307249, A326358, A326363.

a(n) = A306505(n) + 1. - Gus Wiseman, Jul 02 2019

a(7) added by Timothy Yusun, Sep 27 2012
a(8) from Pawelski added by Michel Marcus, Sep 01 2021
a(9) from Pawelski added by Michel Marcus, May 11 2023

A006052 Number of magic squares of order n composed of the numbers from 1 to n^2, counted up to rotations and reflections.

Original entry on oeis.org

1, 0, 1, 880, 275305224, 17753889197660635632

Offset: 1

N. J. A. Sloane

a(4) computed by Frenicle de Bessy (1605? - 1675), published in 1693. The article mentions the 880 squares and considers also 5*5, 6*6, 8*8, and other squares. - Paul Curtz, Jul 13 and Aug 12 2011
a(5) computed by Richard C. Schroeppel in 1973.
According to Pinn and Wieczerkowski, a(6) = (0.17745 +- 0.00016) * 10^20. - R. K. Guy, May 01 2004
a(6) computed by Hidetoshi Mino in 2024 - Hidetoshi Mino, May 31 2024

			An illustration of the unique (up to rotations and reflections) magic square of order 3:
  +---+---+---+
  | 2 | 7 | 6 |
  +---+---+---+
  | 9 | 5 | 1 |
  +---+---+---+
  | 4 | 3 | 8 |
  +---+---+---+

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Vol. II, pp. 778-783 gives the 880 4 X 4 squares.
M. Gardner, Mathematical Games, Sci. Amer. Vol. 249 (No. 1, 1976), p. 118.
M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 216.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ian Cameron, Adam Rogers and Peter Loly, "The Library of Magical Squares" -- a summary of the main results for the Shannon entropy of magic and Latin squares: isentropic clans and indexing, in celebration of George Styan's 75th.
Bernard Frénicle de Bessy, Des carrez ou tables magiques, Divers ouvrages de mathématique et de physique (1693), pp. 423-483.
Bernard Frénicle de Bessy, Table générale des carrez de quatre, Divers ouvrages de mathématique et de physique (1693), pp. 484-503.
Skylar R. Croy, Jeremy A. Hansen, and Daniel J. McQuillan, Calculating the Number of Order-6 Magic Squares with Modular Lifting, Proceedings of the Ninth International Symposium on Combinatorial Search (SoCS 2016).
Mahadi Hasan and Md. Masbaul Alam Polash, An Efficient Constraint-Based Local Search for Maximizing Water Retention on Magic Squares, Emerging Trends in Electrical, Communications, and Information Technologies, Lecture Notes in Electrical Engineering book series (LNEE 2019) Vol. 569, 71-79.
Hidetoshi Mino, The number of magic squares of order 6.
Hidetoshi Mino, Fast enumeration of magic squares, YouTube video, 2025.
I. Peterson, Magic Tesseracts.
K. Pinn and C. Wieczerkowski, Number of magic squares from parallel tempering Monte Carlo, arXiv:cond-mat/9804109 [cond-mat.stat-mech], 1998; Internat. J. Modern Phys., 9 (4) (1998) 541-546.
Tyler Pringle, Magic Squares and Using Magic Series for Theory, The College of William and Mary (2024). See pp. 6, 9.
Artem Ripatti, On the number of semi-magic squares of order 6, arXiv:1807.02983 [math.CO], 2018. See Table 1 p. 2.
R. Schroeppel, Emails to N. J. A. Sloane, Jun. 1991.
N. J. A. Sloane & J. R. Hendricks, Correspondence, 1974.
Walter Trump, How many magic squares are there? - Results of historical and computer enumeration.
Eric Weisstein's World of Mathematics, Magic Square.
Index entries for sequences related to magic squares

Cf. A270876, A271103, A271104.

Definition corrected by Max Alekseyev, Dec 25 2015
a(6) from Hidetoshi Mino, Jul 17 2023
Incorrect a(6) removed by Hidetoshi Mino, Sep 07 2023
a(6) from Hidetoshi Mino, May 31 2024

A006560 Smallest starting prime for n consecutive primes in arithmetic progression.

Original entry on oeis.org

2, 2, 3, 251, 9843019, 121174811

Offset: 1

N. J. A. Sloane

The primes following a(5) and a(6) occur at a(n)+30*k, k=0..(n-1). a(6) was found by Lander and Parkin. The next term requires a spacing >= 210. The expected size is a(7) > 10^21 (see link). - Hugo Pfoertner, Jun 25 2004
From Daniel Forgues, Jan 17 2011: (Start)
It is conjectured that there are arithmetic progressions of n consecutive primes for any n.
Common differences of first and smallest AP of n >= 1 consecutive primes: {0, 1, 2, 6, 30, 30, >= 210, >= 210, >= 210, >= 210, >= 2310, ...} (End)
a(7) <= 71137654873189893604531, found by P. Zimmermann, cf. J. K. Andersen link. - Bert Dobbelaere, Jul 27 2022

			First and smallest occurrence of n, n >= 1, consecutive primes in arithmetic progression:
a(1) = 2: (2) (degenerate arithmetic progression);
a(2) = 2: (2, 3) (degenerate arithmetic progression);
a(3) = 3: (3, 5, 7);
a(4) = 251: (251, 257, 263, 269);
a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);
a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jens Kruse Andersen, The smallest known CPAP-k.
Thomas Bloom, Let k >= 3. Are there k consecutive primes in arithmetic progression?, Erdős Problems.
Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, and P. Zimmermann, Ten consecutive primes in arithmetic progression, Math. Comp., Vol. 71, No. 239 (2002) 1323-1328.
Daniel Forgues, Wiki about consecutive primes in arithmetic progression.
L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp., Vol. 21, No. 99 (1967) p 489.
Terence Tao, Erdős problem database, see no. 141.
Manfred Toplic, The nine and ten primes project, 2004.
Index entries for sequences related to primes in arithmetic progressions

Cf. A005115, A006562, A093364, A126989.
a(5) corresponds to A052243(20) followed by A052243(21) 9843049.
Cf. A089180: indices primes a(n).
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4), A033451: start of CPAP-4 with common difference 6, A052239: start of first CPAP-4 with common difference 6n.
Cf. A059044: start of 5 consecutive primes in arithmetic progression, A210727: CPAP-5 with common difference 60.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.

Join[{2},Table[SelectFirst[Partition[Prime[Range[691*10^4]],n,1], Length[ Union[ Differences[ #]]] == 1&][[1]],{n,2,6}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 10 2019 *)

a(n) = A000040(A089180(n)), or A089180(n) = A000720(a(n)). - M. F. Hasler, Oct 27 2018

Edited by Daniel Forgues, Jan 17 2011

A039951 a(n) is the smallest prime p such that p^2 divides n^(p-1) - 1.

Original entry on oeis.org

2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3

Offset: 1

David W. Wilson

a(n^k) <= a(n) for any n,k > 1.
a(n) is currently unknown for n in {47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, ...}. - Richard Fischer, Jul 15 2021
a(47) > 1.4*10^14, a(72) > 1.4*10^14 (see Fischer's tables).
For all nonnegative integers n and k, a(n^(n^k)) = a(n) (see Puzzle 762 in the links). Also a(n) = 3 if and only if mod(n, 36) is in the set {8, 10, 19, 26, 28, 35}. - Farideh Firoozbakht and Jahangeer Kholdi, Nov 01 2014

C. K. Caldwell, The Prime Glossary, Fermat quotient.
Richard Fischer, Fermat quotients B^(P-1) == 1 (mod P^2)
Richard Fischer, Update Table of n, July 15 2021.
W. Keller and J. Richstein, Fermat quotients q_p(a) that are divisible by p.
Carlos Rivera, Puzzle 762. Conjecture from Ribenboim's book, The Prime Puzzles and Problems Connection.
Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (with missing terms)

Cf. A001220, A045616, A096082, A014127, A123692, A123693, A174422.

Table[p = 2; While[! Divisible[n^(p - 1) - 1, p^2], p = NextPrime@ p]; p, {n, 33}] (* Michael De Vlieger, Nov 24 2016 *)
f[n_] := Block[{p = 2}, While[ PowerMod[n, p - 1, p^2] != 1, p = NextPrime@ p]; p]; Array[f, 33] (* Robert G. Wilson v, Jul 18 2018 *)

a(n)={forprime(p=2, oo, if(Mod(n, p^2)^(p-1)==1, return(p))); oo} \\ Felix Fröhlich, Jul 24 2014

a(4k+1) = 2.

a(n) = A096082(n) for all n > 1 that are not of the form 4k+1. Note that A096082 begins with n = 2. [Corrected and clarified by Jonathan Sondow, Jun 17-18 2010]

a(34)-a(46) from Helmut Richter (richter(AT)lrz.de), May 17 2004
Entry revised by N. J. A. Sloane, Nov 30 2006
Edited by Max Alekseyev, Oct 06, Oct 09 2009
Edited and updated by Max Alekseyev, Jan 29 2012

cp's OEIS Frontend

A102896 Number of ACI algebras (or semilattices) on n generators with no annihilator.

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A003182 Dedekind numbers: inequivalent monotone Boolean functions of n or fewer variables, or antichains of subsets of an n-set.

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