keyword:hard - cp's OEIS frontend

A111441 Numbers k such that the sum of the squares of the first k primes is divisible by k.

1, 19, 37, 455, 509, 575, 20597, 202717, 1864637, 542474231, 1139733677, 51283502951, 230026580777, 22148897608321, 51271840444039, 1820988137264459

Offset: 1

Stefan Steinerberger, Nov 14 2005

a(16) > 10^14 if it exists. - Anders Kaseorg, Dec 02 2020
Conjecture: There are no terms that are 3 or 9 modulo 12. This seems to hold for all related sequences with even powers of primes, not just squares. Compare "sums of powers of primes divisibility sequences", linked below. - Daniel Bamberger, Dec 03 2020
From Jacob Christian Munch-Andersen, Dec 13 2020: (Start)
Any prime except 3 raised to the 2nd power is 1 modulo 3. Therefore adding the squared primes together results in a simple periodic pattern modulo 3. Any term that is 0 modulo 3 would imply that it divides a number that is 2 modulo 3; as this is impossible there cannot be any terms divisible by 3.
The same proof indeed holds for similar lists generated with any even power, and a similar proof for instance disqualifies any multiple of 5 from the similar 4th-power list. A slightly simpler similar proof shows that there are no terms divisible by 2.
(End)
The previous comment implies that for a list generated with the m-th power, there are no terms divisible by p when p is prime and p-1 is a divisor of m. For example, the 12th power list has no terms divisible by 2, 3, 5, 7 or 13. - Paul W. Dyson, Jan 09 2021
The periodic pattern of the sum of primes raised to an even power as described in the comments above follows from Fermat's little theorem. When the pattern is periodic for a given p it can be seen that when k mod p = 0 the sum mod p = p-1 and therefore sum mod k cannot be 0. - Bruce Garner, Apr 08 2021
a(2) is also a value in each of the lists generated with the powers 20, 38, 56... . a(3) is also a value in each of the lists generated with the powers 38, 74, 110... . In general, if the sum of the first k primes each to the power of m is divisible by k, and m >= the maximum exponent in the prime factorization of k, then the sum of the first k primes each to the power of m + j * psi(k) is also divisible by k, where psi(k) is the reduced totient function (A002322) and j is any positive integer. This follows from the fact that n^m == n^(m + psi(k)) (mod k) for all integers n and all integers m >= the maximum exponent in the prime factorization of k. - Paul W. Dyson, Dec 09 2022
a(17) > 8*10^15. - Paul W. Dyson, Jan 16 2025

			The sum of the squares of the first 19 primes 2^2 + 3^2 + 5^2 + ... + 67^2 = 19*1314, thus 19 is in the sequence.

OEIS Wiki, Sums of powers of primes divisibility sequences.
Matt Parker, MPMSolutions: The 19 Challenge, YouTube video, 2020 (challenge asking for terms of this sequence).
Wikipedia, Fermat's little theorem

Cf. A024450, A045345.

Cf. also A217599, A217600 for the corresponding prime numbers and sums.

s = 0; t = {}; Do[s = s + Prime[n]^2; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 10^6}]; t (* Robert G. Wilson v, Nov 15 2005 *)
Module[{nn=2*10^6,pr2},pr2=Accumulate[Prime[Range[nn]]^2];Select[Thread[{Range[nn],pr2}],Divisible[#[[2]],#[[1]]]&]][[;;,1]] (* The program generates the first 9 terms of the sequence. *) (* Harvey P. Dale, May 25 2025 *)

a := 0; for n from 1 to 100000 do a := a + ithprime(n)^2; if a/n = trunc(a/n) then print(n); end_if; end_for;

for(n=1, 2*10^11, m=n; s=0; while(m>0, s=s+prime(m)^2; m--); if(s%n==0, print1(n, ", "))) \\ Felix Fröhlich, Jul 07 2014

isok(n) = norml2(primes(n)) % n == 0; \\ Michel Marcus, Nov 25 2020

a(8)-a(9) from Robert G. Wilson v, Nov 15 2005
a(10)-a(11) from Ryan Propper, Mar 27 2007
a(12) from Robert Price, Mar 19 2013
a(13) from Balázs Dura-Kovács, Nov 25 2020
a(14) from Balázs Dura-Kovács, Nov 30 2020
a(15) from Anders Kaseorg, Dec 02 2020
a(16) from Jonas Lippuner, Aug 23 2021

A128165 Numbers k such that k divides 1 plus the sum of the first k primes.

Original entry on oeis.org

1, 2, 6, 10, 20, 22, 28, 155, 488, 664, 992, 6162, 7840, 7975, 8793, 18961, 32422, 148220, 231625, 332198, 459121, 462932, 2115894, 8108930, 10336641, 11789731, 15500046, 23483195, 46571611, 48582404, 77033887, 105390951, 132421841, 229481560, 1224959312

Offset: 1

Alexander Adamchuk, Feb 22 2007

a(44) > 4.4*10^10. - Robert Price, Dec 15 2013

a(50) > 10^14. - Bruce Garner, Jun 05 2021

Bruce Garner, Table of n, a(n) for n = 1..49
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).

Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248, A024450, A111441, A217599, A128166, A233862, A217600, A217601.

k = 0; s = 1; p = 2; A128165 = {}; While[k < 247336000, If[Mod[s += p, ++k] == 0, AppendTo[A128165, k]; Print[{k, p}]]; p = NextPrime@ p]; A128165

is(n)=sum(i=1,n,prime(i),1)%n==0 \\ Charles R Greathouse IV, Nov 07 2014

n=0; s=1; forprime(p=2,1e9, s+=p; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Nov 07 2014

More terms from Ryan Propper, Apr 05 2007
a(34) from Robert G. Wilson v, Jan 21 2011
a(35) from Robert Price, Dec 15 2013

A128166 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^2 = 1 + A024450(k).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 13, 26, 28, 45, 66, 174, 308, 350, 366, 417, 783, 804, 3774, 5714, 7998, 17628, 17940, 63447, 67620, 83028, 137868, 216717, 297486, 425708, 659316, 674166, 883500, 1203786, 3605052, 6778607, 9516098, 19964862, 25338586, 27771732, 70980884, 91871891, 208234138, 231967260, 238066596, 289829748, 784027092, 1078515812, 33256634230

Offset: 1

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

a(51) > 5.3*10^10. - Robert Price, Dec 16 2013

a(67) > 7*10^13. - Bruce Garner, May 05 2021

Bruce Garner, Table of n, a(n) for n = 1..66
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

s = 1; Do[s = s + Prime[n]^2; If[ Mod[s, n] == 0, Print[n]], {n, 700000}]
(* or *)
Select[Range[10^4], IntegerQ[(1 + Plus@@(Prime[Range[#]]^2))/#] &] (* Alonso del Arte, Jan 20 2011 *)

More terms from Sean A. Irvine, Jan 20 2011

a(45)-a(50) from Robert Price, Dec 16 2013

A000170 Number of ways of placing n nonattacking queens on an n X n board.

Original entry on oeis.org

1, 1, 0, 0, 2, 10, 4, 40, 92, 352, 724, 2680, 14200, 73712, 365596, 2279184, 14772512, 95815104, 666090624, 4968057848, 39029188884, 314666222712, 2691008701644, 24233937684440, 227514171973736, 2207893435808352, 22317699616364044, 234907967154122528

Offset: 0

N. J. A. Sloane

For n > 3, a(n) is the number of maximum independent vertex sets in the n X n queen graph. - Eric W. Weisstein, Jun 20 2017
Number of nodes on level n of the backtrack tree for the n queens problem (a(n) = A319284(n, n)). - Peter Luschny, Sep 18 2018
Number of permutations of [1...n] such that |p(j)-p(i)| != j-i for iXiangyu Chen, Dec 24 2020
M. Simkin shows that the number of ways to place n mutually nonattacking queens on an n X n chessboard is ((1 +/- o(1))*n*exp(-c))^n, where c = 1.942 +/- 0.003. These are approximately (0.143*n)^n configurations. - Peter Luschny, Oct 07 2021

			a(2) = a(3) = 0, since on 2 X 2 and 3 X 3 chessboards there are no solutions.
.
a(4) = 2:
  +---------+ +---------+
  | . . Q . | | . Q . . |
  | Q . . . | | . . . Q |
  | . . . Q | | Q . . . |
  | . Q . . | | . . Q . |
  +---------+ +---------+
a(5) = 10:
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
  | . . . Q . | | . . Q . . | | . . . . Q | | . . . Q . | | . . . . Q |
  | . Q . . . | | . . . . Q | | . . Q . . | | Q . . . . | | . Q . . . |
  | . . . . Q | | . Q . . . | | Q . . . . | | . . Q . . | | . . . Q . |
  | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | Q . . . . |
  | Q . . . . | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
  | Q . . . . | | . Q . . . | | Q . . . . | | . . Q . . | | . Q . . . |
  | . . . Q . | | . . . . Q | | . . Q . . | | Q . . . . | | . . . Q . |
  | . Q . . . | | . . Q . . | | . . . . Q | | . . . Q . | | Q . . . . |
  | . . . . Q | | Q . . . . | | . Q . . . | | . Q . . . | | . . Q . . |
  | . . Q . . | | . . . Q . | | . . . Q . | | . . . . Q | | . . . . Q |
  +-----------+ +-----------+ +-----------+ +-----------+ +-----------+
a(6) = 4:
  +-------------+ +-------------+ +-------------+ +-------------+
  | . . . . Q . | | . . . Q . . | | . . Q . . . | | . Q . . . . |
  | . . Q . . . | | Q . . . . . | | . . . . . Q | | . . . Q . . |
  | Q . . . . . | | . . . . Q . | | . Q . . . . | | . . . . . Q |
  | . . . . . Q | | . Q . . . . | | . . . . Q . | | Q . . . . . |
  | . . . Q . . | | . . . . . Q | | Q . . . . . | | . . Q . . . |
  | . Q . . . . | | . . Q . . . | | . . . Q . . | | . . . . Q . |
  +-------------+ +-------------+ +-------------+ +-------------+
- _Hugo Pfoertner_, Mar 17 2019

M. Gardner, The Unexpected Hanging, pp. 190-2, Simon & Shuster NY 1969
Jieh Hsiang, Yuh-Pyng Shieh and Yao-Chiang Chen, The cyclic complete mappings counting problems, in Problems and Problem Sets for ATP, volume 02-10 of DIKU technical reports, G. Sutcliffe, J. Pelletier and C. Suttner, eds., 2002.
D. E. Knuth, The Art of Computer Programming, Volume 4, Pre-fascicle 5B, Introduction to Backtracking, 7.2.2. Backtrack programming. 2018.
M. Kraitchik, The Problem of The Queens, Mathematical Recreations, 2nd ed., New York, Dover, 1953, pp. 247-256.
Massimo Nocentini, "An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation", PhD Thesis, University of Florence, 2019. See Ex. 67.
W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York, Dover, 1987, pp. 166-172 (The Eight Queens Problem).
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
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).
R. J. Walker, An enumerative technique for a class of combinatorial problems, pp. 91-94 of Proc. Sympos. Applied Math., vol. 10, Amer. Math. Soc., 1960.
M. B. Wells, Elements of Combinatorial Computing. Pergamon, Oxford, 1971, p. 238.

Jordan Bell and Brett Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics, Volume 309, Issue 1, Jan 06 2009, Pages 1-31.
D. Bill, Durango Bill's The N-Queens Problem
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656.
J. R. Bitner and E. M. Reingold, Backtrack programming techniques, Commun. ACM, 18 (1975), 651-656. [Annotated scanned copy]
Candida Bowtell and Peter Keevash, The n-queens problem, arXiv:2109.08083 [math.CO] 2021.
P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
V. Chvatal, All solutions to the problem of eight queens
V. Chvatal, All solutions to the problem of eight queens [Cached copy, pdf format, with permission]
Gheorghe Coserea, Solutions for n=8.
Gheorghe Coserea, Solutions for n=9.
Gheorghe Coserea, MiniZinc model for generating solutions.
Matteo Fischetti and Domenico Salvagnin, Chasing First Queens by Integer Programming, 2018.
Matteo Fischetti and Domenico Salvagnin, Finding First and Most-Beautiful Queens by Integer Programming, arXiv:1907.08246 [cs.DS], 2019.
J. Freeman, A neural network solution to the n-queens problem, The Mathematica J., 3 (No. 3, 1993), 52-56.
Ian P. Gent, Christopher Jefferson and Peter Nightingale, Complexity of n-Queens Completion, Journal of Artificial Intelligence Research 59 (2017), see p 816.
Eric Grigoryan, Investigation of the Regularities in the Formation of Solutions n-Queens Problem, Modeling of Artificial Intelligence, 2018, 5(1), 3-21.
E. Grigoryan, Linear algorithm for solution n-Queens Completion problem, arXiv:1912.05935 [cs.AI], 2019.
James Grime and Brady Haran, The 8 Queen Problem, Numberphile video (2015).
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.
Kenji Kise, Takahiro Katagiri, Hiroki Honda and Toshitsugu Yuba, Solving the 24-queens Problem using MPI on a PC Cluster, Technical Report UEC-IS-2004-6, Graduate School of Information Systems, The University of Electro-Communications (2004).
D. E. Knuth, Donald Knuth's 24th Annual Christmas Lecture: Dancing Links, Stanfordonline, Video published on YouTube, Dec 12, 2018.
W. Kosters, n-Queens bibliography
Vaclav Kotesovec, Non-attacking chess pieces, Sixth edition, 795 pages, Feb 02 2013 (minor update Mar 29 2016).
Zur Luria, New bounds on the number of n-queens configurations, arXiv:1705.05225 [math.CO], 2017.
Zur Luria, Michael Simkin, A lower bound for the n-queens problem, arXiv:2105.11431 [math.CO], 2021.
E. Masehian, H. Akbaripour and N. Mohabbati-Kalejahi, Solving the n Queens Problem using a Tuned Hybrid Imperialist Competitive Algorithm, 2013.
E. Masehian, H. Akbaripour and N. Mohabbati-Kalejahi, Landscape analysis and efficient metaheuristics for solving the n-queens problem, Computational Optimization and Applications, 2013; DOI 10.1007/s10589-013-9578-z.
Nasrin Mohabbati-Kalejahi, Hossein Akbaripour and Ellips Masehian, Basic and Hybrid Imperialist Competitive Algorithms for Solving the Non-attacking and Non-dominating n -Queens Problems, Studies in Computational Intelligence Volume 577, 2015, pp 79-96. DOI 10.1007/978-3-319-11271-8_6.
Ralph Morrison and Noah Speeter, The Gonality of Queen's Graphs, arXiv:2312.04686 [math.CO], 2023.
Parth Nobel, Akshay Agrawal and Stephen Boyd, Computing tighter bounds on the n-queens constant via Newton’s method, arXiv:2112.03336 [math.CO], 2021.
J. Pope and D. Sonnier, A linear solution to the n-Queens problem using vector spaces, Journal of Computing Sciences in Colleges, Volume 29 Issue 5, May 2014 Pages 77-83.
T. B. Preußer and M. R. Engelhardt, Putting Queens in Carry Chains, No. 27, Journal of Signal Processing Systems, Volume 88, Issue 2, August 2017. (The title refers to the fact that the article discusses the case n = 27.)
Thomas B. Preußer, Bernd Nägel and Rainer G. Spallek, Putting Queens in Carry Chains, Slides, HIPEAC WRC'09.
E. M. Reingold, Letter to N. J. A. Sloane, Dec 27 1973
I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
Werner Florian Samayoa, Maria Liz Crespo, Sergio Carrato, Agustin Silva, and Andres Cicuttin, HyperFPGA: An Experimental Testbed for Heterogeneous Supercomputing, 2023.
Michael Simkin, The number of n-queens configurations, arXiv:2107.13460 [math.CO], 2021.
Wenxi Wang, Muhammad Usman, Alyas Almaawi, Kaiyuan Wang, Kuldeep S. Meel and Sarfraz Khurshid, A Study of Symmetry Breaking Predicates and Model Counting, National University of Singapore (2020).
Eric Weisstein's World of Mathematics, Maximum Independent Vertex Set
Eric Weisstein's World of Mathematics, Queen Graph
Eric Weisstein's World of Mathematics, Queens Problem
M. B. Wells, Elements of Combinatorial Computing, Pergamon, Oxford, 1971. [Annotated scanned copy of pages 237-240]
Wikipedia, Eight Queens Puzzle
Cheng Zhang and Jianpeng Ma, Counting Solutions for the N-queens and Latin Square Problems by Efficient Monte Carlo Simulations, arXiv:0808.4003 [cond-mat.stat-mech], 2008.

Cf. A002562, A065256.
Cf. A036464 (2Q), A047659 (3Q), A061994 (4Q), A108792 (5Q), A176186 (6Q).
Cf. A099152, A006717, A051906, A319284 (backtrack trees).
Cf. A260318, A260319, A260320.
Main diagonal of A348129.

Strong conjecture: there is a constant c around 2.54 such that a(n) is asymptotic to n!/c^n; weak conjecture: lim_{n -> infinity} (1/n) * log(n!/a(n)) = constant = 0.90.... - Benoit Cloitre, Nov 10 2002
Lim_{n->infinity} a(n)^(1/n)/n = exp(-A359441) = 0.1431301... [Simkin 2021]. - Vaclav Kotesovec, Jan 01 2023
a(n) = 8 * A260320(n) + 4 * A260319(n) + 2 * A260318(n) for n >= 2 (see Kraitchik reference). - Jason Bard, Aug 12 2025

Terms for n=21-23 computed by Sylvain PION (Sylvain.Pion(AT)sophia.inria.fr) and Joel-Yann FOURRE (Joel-Yann.Fourre(AT)ens.fr).
a(24) from Kenji KISE (kis(AT)is.uec.ac.jp), Sep 01 2004
a(25) from Objectweb ProActive INRIA Team (proactive(AT)objectweb.org), Jun 11 2005 [Communicated by Alexandre Di Costanzo (Alexandre.Di_Costanzo(AT)sophia.inria.fr)]. This calculation took about 53 years of CPU time.
a(25) has been confirmed by the NTU 25Queen Project at National Taiwan University and Ming Chuan University, led by Yuh-Pyng (Arping) Shieh, Jul 26 2005. This computation took 26613 days CPU time.
The NQueens-at-Home web site gives a different value for a(24), 226732487925864. Thanks to Goran Fagerstrom for pointing this out. I do not know which value is correct. I have therefore created a new entry, A140393, which gives the NQueens-at-home version of the sequence. - N. J. A. Sloane, Jun 18 2008
It now appears that this sequence (A000170) is correct and A140393 is wrong. - N. J. A. Sloane, Nov 08 2008
Added a(26) as calculated by Queens(AT)TUD [http://queens.inf.tu-dresden.de/]. - Thomas B. Preußer, Jul 11 2009
Added a(27) as calculated by the Q27 Project [https://github.com/preusser/q27]. - Thomas B. Preußer, Sep 23 2016
a(0) = 1 prepended by Joerg Arndt, Sep 16 2018

A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

Original entry on oeis.org

1, 1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663, 228779330204, 1758309223457, 13573319825615, 105192814197984, 818136047201932, 6383528588447574

Offset: 1

N. J. A. Sloane

a(1)-a(12) computed by Achim Flammenkamp.
A000162 but with one copy of each mirror-image deleted.
From R. J. Mathar, Mar 19 2018: (Start)
We can split the numbers into an irregular table which lists in row n how many configurations have c contacts for c >= 0:
  1;
  0 1;
  0 0 2;
  0 0 0 6 1;
  0 0 0 0 21 2;
  0 0 0 0 0 91 19 2;
  0 0 0 0 0 0 484 110 12 1;
  0 0 0 0 0 0 0 2817 852 129 12 0 1;
  0 0 0 0 0 0 0 0 17788 6321 1166 132 5 1;
Row lengths are 1+A007818(n). Row sums are a(n).
(End)
Number of unoriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For unoriented polyominoes, chiral pairs are counted as one.- Robert A. Russell, Mar 21 2024

S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972. [See https://books.google.nl/books?id=ja7iBQAAQBAJ&pg=PA101]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Achim Flammenkamp, Home page.
Kevin L. Gong, Polyominoes Home Page.
John Mason, Counting free polycubes.
John Mason, Coordinate sets of examples of polycube symmetry (version 2)

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.
32nd row of A366766.
Cf. A000162 (oriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).
Cf. for each symmetry: A376964, A376965, A376966, A376967, A376968, A376969, A376970, A376972, A376973, A376974, A376975, A376976, A376977, A376978, A376979, A376980, A376981, A376982, A376983, A377127, A376984, A376985, A376986, A376987, A376988, A376989, A377128, A376990, A376991, A377129, A377130, A377131, A376971

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000162 = A@000162;
A007743 = A@007743;
a[n_] := (A007743[[n]] + A000162[[n]])/2;
a /@ Range[16] (* Jean-François Alcover, Jan 16 2020 *)

a(n) = A000162(n) - A371397(n) = A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Jan 02 2002
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
More terms from John Mason, Sep 19 2024

A062666 Numbers k such that 100^k - 99^k is prime.

Original entry on oeis.org

2, 5, 19, 59, 1013, 2371, 13967, 44683

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms > 10000 correspond to probable primes.

a(9) > 10^5. - Robert Price, Jul 10 2013

Cf. A000043, A057468, A059801, A059802, A062572-A062665.

lst={}; k=100; Do[If[PrimeQ[k^n-(k-1)^n], Print[n]; AppendTo[lst, n]], {n, 10^5}]; lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

forprime(p=2,1e5,if(ispseudoprime(100^p-99^p),print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

Edited by T. D. Noe, Oct 30 2008

a(7)-a(8) from Robert Price, Jul 10 2013

A002982 Numbers k such that k! - 1 is prime.

Original entry on oeis.org

3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480, 34790, 94550, 103040, 147855, 208003

Offset: 1

N. J. A. Sloane

The corresponding primes n!-1 are often called factorial primes.

			From _Gus Wiseman_, Jul 04 2019: (Start)
The sequence of numbers n! - 1 together with their prime indices begins:
                    1: {}
                    5: {3}
                   23: {9}
                  119: {4,7}
                  719: {128}
                 5039: {675}
                40319: {9,273}
               362879: {5,5,430}
              3628799: {10,11746}
             39916799: {6,7,9,992}
            479001599: {25306287}
           6227020799: {270,256263}
          87178291199: {3610490805}
        1307674367999: {7,11,11,16,114905}
       20922789887999: {436,318519035}
      355687428095999: {8,21,10165484947}
     6402373705727999: {17,20157,25293727}
   121645100408831999: {119,175195,4567455}
  2432902008176639999: {11715,659539127675}
(End)

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 166, p. 53, Ellipses, Paris 2008.
R. K. Guy, Unsolved Problems in Number Theory, Section A2.
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 118.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 719 at p. 160.

A. Borning, Some results for k!+-1 and 2.3.5...p+-1, Math. Comp., 26:118 (1972), pp. 567-570.
J. P. Buhler et al., Primes of the form n!+-1 and 2.3.5....p+-1, Math. Comp., 38:158 (1982), pp. 639-643.
Chris K. Caldwell, Factorial Primes.
C. K. Caldwell and Y. Gallot, On the primality of n!+-1 and 2*3*5*...*p+-1, Math. Comp., 71:237 (2002), pp. 441-448.
P. Carmody, Factorial Prime Search Progress Pages.
Antonín Čejchan, Michal Křížek, and Lawrence Somer, On Remarkable Properties of Primes Near Factorials and Primorials, Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4.
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.
R. K. Guy and N. J. A. Sloane, Correspondence, 1985.
H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3, 1987), 197-203. (Annotated scanned copy)
Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, 99 (2015), pp 213-219. doi:10.1017/mag.2015.28.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012.
R. Ondrejka, The Top Ten: a Catalogue of Primal Configurations.
PrimeGrid, World Record Factorial Prime!!!.
PrimeGrid, Announcement of 94550, (2010). - _Felix Fröhlich_, Jul 11 2014
PrimeGrid, Announcement of 103040, (2010). - _Felix Fröhlich_, Jul 11 2014
PrimeGrid, Announcement of 147855, (2013). - _Felix Fröhlich_, Jul 11 2014
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Factorial.
Eric Weisstein's World of Mathematics, Factorial Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1989.
Index entries for sequences related to factorial numbers.

Cf. A002981 (numbers n such that n!+1 is prime).
Cf. A055490 (primes of form n!-1).
Cf. A088332 (primes of form n!+1).
Cf. A000142, A001221, A001222, A046051, A054991, A112798, A325272.

[n: n in [0..500] | IsPrime(Factorial(n)-1)]; // Vincenzo Librandi, Sep 07 2017

Select[Range[10^3], PrimeQ[ #!-1] &] (* Vladimir Joseph Stephan Orlovsky, May 01 2008 *)

is(n)=ispseudoprime(n!-1) \\ Charles R Greathouse IV, Mar 21 2013

from sympy import factorial, isprime
A002982_list = [n for n in range(1,10**2) if isprime(factorial(n)-1)] # Chai Wah Wu, Apr 04 2021

21480 sent in by Ken Davis (ken.davis(AT)softwareag.com), Oct 29 2001
Updated Feb 26 2007 by Max Alekseyev, based on progress reported in the Carmody web site.
Inserted missing 21480 and 34790 (see Caldwell). Added 94550, discovered Oct 05 2010. Eric W. Weisstein, Oct 06 2010
103040 was discovered by James Winskill, Dec 14 2010. It has 471794 digits. Corrected by Jens Kruse Andersen, Mar 22 2011
a(26) = 147855 from Felix Fröhlich, Sep 02 2013
a(27) = 208003 from Sou Fukui, Jul 27 2016

A000798 Number of different quasi-orders (or topologies, or transitive digraphs) with n labeled elements.

Original entry on oeis.org

1, 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021, 207881393656668953041, 115617051977054267807460, 88736269118586244492485121, 93411113411710039565210494095, 134137950093337880672321868725846, 261492535743634374805066126901117203

Offset: 0

N. J. A. Sloane

From Altug Alkan, Dec 18 2015 and Feb 28 2017: (Start)
a(p^k) == k+1 (mod p) for all primes p. This is proved by Kizmaz at On The Number Of Topologies On A Finite Set link. For proof see Theorem 2.4 in page 2 and 3. So a(19) == 2 (mod 19).
a(p+n) == A265042(n) (mod p) for all primes p. This is also proved by Kizmaz at related link, see Theorem 2.7 in page 4. If n=2 and p=17, a(17+2) == A265042(2) (mod 17), that is a(19) == 51 (mod 17). So a(19) is divisible by 17.
In conclusion, a(19) is a number of the form 323*n - 17. (End)
The BII-numbers of finite topologies without their empty set are given by A326876. - Gus Wiseman, Aug 01 2019
From Tian Vlasic, Feb 23 2022: (Start)
Although no general formula is known for a(n), by considering the number of topologies with a fixed number of open sets, it is possible to explicitly represent the sequence in terms of Stirling numbers of the second kind.
For example: a(n,3) = 2*S(n,2), a(n,4) = S(n,2) + 6*S(n,3), a(n,5) = 6*S(n,3) + 24*S(n,4).
Lower and upper bounds are known: 2^n <= a(n) <= 2^(n*(n-1)), n > 1.
This follows from the fact that there are 2^(n*(n-1)) reflexive relations on a set with n elements.
Furthermore: a(n+1) <= a(n)*(3a(n)+1). (End)

			From _Gus Wiseman_, Aug 01 2019: (Start)
The a(3) = 29 topologies are the following (empty sets not shown):
  {123}  {1}{123}   {1}{12}{123}  {1}{2}{12}{123}   {1}{2}{12}{13}{123}
         {2}{123}   {1}{13}{123}  {1}{3}{13}{123}   {1}{2}{12}{23}{123}
         {3}{123}   {1}{23}{123}  {2}{3}{23}{123}   {1}{3}{12}{13}{123}
         {12}{123}  {2}{12}{123}  {1}{12}{13}{123}  {1}{3}{13}{23}{123}
         {13}{123}  {2}{13}{123}  {2}{12}{23}{123}  {2}{3}{12}{23}{123}
         {23}{123}  {2}{23}{123}  {3}{13}{23}{123}  {2}{3}{13}{23}{123}
                    {3}{12}{123}
                    {3}{13}{123}        {1}{2}{3}{12}{13}{23}{123}
                    {3}{23}{123}
(End)

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.
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 229.
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 243.
Levinson, H.; Silverman, R. Topologies on finite sets. II. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 699--712, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561090 (81c:54006)
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).
For further references concerning the enumeration of topologies and posets see under A001035.
G.H. Patil and M.S. Chaudhary, A recursive determination of topologies on finite sets, Indian Journal of Pure and Applied Mathematics, 26, No. 2 (1995), 143-148.

V. I. Arnautov and A. V. Kochina, Method for constructing one-point expansions of a topology on a finite set and its applications, Bul. Acad. Stiinte Republ. Moldav. Matem. 3 (64) (2010) 67-76.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Moussa Benoumhani and Ali Jaballah, Chains in lattices of mappings and finite fuzzy topological spaces, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 99-111.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5.
Juliana Bowles and Marco B. Caminati, A Verified Algorithm Enumerating Event Structures, arXiv:1705.07228 [cs.LO], 2017.
Gunnar Brinkmann and Brendan D. McKay, Posets on up to 16 points.
G. Brinkmann and B. D. McKay, Posets on up to 16 Points, Order 19 (2) (2002) 147-179 (Table IV).
J. I. Brown and S. Watson, The number of complements of a topology on n points is at least 2^n (except for some special cases), Discr. Math., 154 (1996), 27-39.
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]
S. D. Chatterji, The number of topologies on n points, Manuscript, 1966 [Annotated scanned copy]
Tyler Clark and Tom Richmond, The Number of Convex Topologies on a Finite Totally Ordered Set, 2013, Involve, Vol. 8 (2015), No. 1, 25-32.
E. D. Cooper, Representation and generation of finite partially ordered sets, Manuscript, no date [Annotated scanned copy]
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259.
M. Erné, Struktur- und Anzahlformeln für Topologien auf Endlichen Mengen, Manuscripta Math., 11 (1974), 221-259. (Annotated scanned copy)
M. Erné and K. Stege, The number of partially ordered (labeled) sets, Preprint, 1989. (Annotated scanned copy)
M. Erné and K. Stege, Counting Finite Posets and Topologies, Order, 8 (1991), 247-265.
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313. [Annotated scanned copy]
J. W. Evans, F. Harary and M. S. Lynn, On the computer enumeration of finite topologies, Commun. ACM, 10 (1967), 295-297, 313.
S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]
Loic Foissy, Claudia Malvenuto, and Frederic Patras, B_infinity-algebras, their enveloping algebras, and finite spaces, arXiv preprint arXiv:1403.7488 [math.AT], 2014.
Loic Foissy, Claudia Malvenuto, and 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. .
L. Foissy and C. Malvenuto, The Hopf algebra of finite topologies and T-partitions, arXiv preprint arXiv:1407.0476 [math.RA], 2014.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
E. N. Gilbert, A catalog of partially ordered systems, unpublished memorandum, Aug 08, 1961. [Annotated scanned copy]
S. Giraudo, J.-G. Luque, L. Mignot and F. Nicart, Operads, quasiorders and regular languages, arXiv preprint arXiv:1401.2010 [cs.FL], 2014.
D. J. Greenhoe, Properties of distance spaces with power triangle inequalities, ResearchGate, 2015.
J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17 (2000) no. 4, 333-341.
Institut f. Mathematik, Univ. Hanover, Erne/Heitzig/Reinhold papers
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
Dongseok Kim, Young Soo Kwon and Jaeun Lee, Enumerations of finite topologies associated with a finite graph, arXiv preprint arXiv:1206.0550[math.CO], 2012.
M. Y. Kizmaz, On The Number Of Topologies On A Finite Set, arXiv preprint arXiv:1503.08359 [math.NT], 2015-2019.
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. J. Kleitman and B. L. Rothschild, The number of finite topologies, Proc. Amer. Math. Soc., 25 (1970), 276-282.
Messaoud Kolli, Direct and Elementary Approach to Enumerate Topologies on a Finite Set, J. Integer Sequences, Volume 10, 2007, Article 07.3.1.
Messaoud Kolli, On the cardinality of the T_0-topologies on a finite set, International Journal of Combinatorics, Volume 2014 (2014), Article ID 798074, 7 pages.
Sami Lazaar, Houssem Sabri, and Randa Tahri, Structural and Numerical Studies of Some Topological Properties for Alexandroff Spaces, Bull. Iran. Math. Soc. (2021).
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
M. Rayburn, On the Borel fields of a finite set, Proc. Amer. Math.. Soc., 19 (1968), 885-889. [Annotated scanned copy]
M. Rayburn and N. J. A. Sloane, Correspondence, 1974
D. Rusin, More info and references [Broken link]
D. Rusin, More info and references [Cached copy]
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198. [Annotated scanned copy]
A. Shafaat, On the number of topologies definable for a finite set, J. Austral. Math. Soc., 8 (1968), 194-198.
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.
Ivo Terek, Smooth Manifolds, Williams College (2025). See p. 6.
Wietske Visser, Koen V. Hindriks and Catholijn M. Jonker, Goal-based Qualitative Preference Systems, 2012.
N. L. White, Two letters to N. J. A. Sloane, 1970, with hand-drawn enclosure
J. A. Wright, Letter to N. J. A. Sloane, Nov 21 1970, with four enclosures
J. A. Wright, There are 718 6-point topologies, quasiorderings and transgraphs, Preprint, 1970 [Annotated scanned copy]
J. A. Wright, Two related abstracts, 1970 and 1972 [Annotated scanned copies]
J. A. Wright, Letter to N. J. A. Sloane, Apr 06 1972, listing 18 sequences
Index entries for "core" sequences

Row sums of A326882.
Cf. A001035 (labeled posets), A001930 (unlabeled topologies), A000112 (unlabeled posets), A006057.
Sequences in the Erné (1974) paper: A000798, A001035, A006056, A006057, A001929, A001927, A006058, A006059, A000110.
Cf. A102894, A102895, A102897, A306445, A326866, A326876, A326878, A326881.

Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&SubsetQ[#,Union[Union@@@Tuples[#,2],DeleteCases[Intersection@@@Tuples[#,2],{}]]]&]],{n,0,3}] (* Gus Wiseman, Aug 01 2019 *)

a(n) = Sum_{k=0..n} Stirling2(n, k)*A001035(k).
E.g.f.: A(exp(x) - 1) where A(x) is the e.g.f. for A001035. - Geoffrey Critzer, Jul 28 2014
It is known that log_2(a(n)) ~ n^2/4. - Tian Vlasic, Feb 23 2022

Two more terms from Jobst Heitzig (heitzig(AT)math.uni-hannover.de), Jul 03 2000

a(17)-a(18) are from Brinkmann's and McKay's paper. - Vladeta Jovovic, Jun 10 2007

A014466 Dedekind numbers: monotone Boolean functions, or nonempty antichains of subsets of an n-set.

Original entry on oeis.org

1, 2, 5, 19, 167, 7580, 7828353, 2414682040997, 56130437228687557907787, 286386577668298411128469151667598498812365

Offset: 0

N. J. A. Sloane

A monotone Boolean function is an increasing functions from P(S), the set of subsets of S, to {0,1}.
The count of antichains includes the antichain consisting of only the empty set, but excludes the empty antichain.
Also counts bases of hereditary systems.
Also antichains of nonempty subsets of an n-set. The unlabeled case is A306505. The spanning case is A307249. This sequence has a similar description to A305000 except that the singletons must be disjoint from the other edges. - Gus Wiseman, Feb 20 2019
a(n) is the total number of hierarchical log-linear models on n labeled factors (categorical variables). See Wickramasinghe (2008) and Nardi and Rinaldo (2012). - Petros Hadjicostas, Apr 08 2020
From Lorenzo Sauras Altuzarra, Apr 02 2023: (Start)
a(n) is the number of labeled abstract simplicial complexes on n vertices.
A058673(n) <= a(n) <= A058891(n+1). (End)

			a(2)=5 from the antichains {{}}, {{1}}, {{2}}, {{1,2}}, {{1},{2}}.
From _Gus Wiseman_, Feb 20 2019: (Start)
The a(0) = 1 through a(3) = 19 antichains:
  {{}}  {{}}   {{}}      {{}}
        {{1}}  {{1}}     {{1}}
               {{2}}     {{2}}
               {{12}}    {{3}}
               {{1}{2}}  {{12}}
                         {{13}}
                         {{23}}
                         {{123}}
                         {{1}{2}}
                         {{1}{3}}
                         {{2}{3}}
                         {{1}{23}}
                         {{2}{13}}
                         {{3}{12}}
                         {{12}{13}}
                         {{12}{23}}
                         {{13}{23}}
                         {{1}{2}{3}}
                         {{12}{13}{23}}
(End)
From _Lorenzo Sauras Altuzarra_, Apr 02 2023: (Start)
The 19 sets E such that ({1, 2, 3}, E) is an abstract simplicial complex:
  {}
  {{1}}
  {{2}}
  {{3}}
  {{1}, {2}}
  {{1}, {3}}
  {{2}, {3}}
  {{1}, {2}, {3}}
  {{1}, {2}, {1, 2}}
  {{1}, {3}, {1, 3}}
  {{2}, {3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}}
  {{1}, {2}, {3}, {1, 3}}
  {{1}, {2}, {3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}}
  {{1}, {2}, {3}, {1, 2}, {2, 3}}
  {{1}, {2}, {3}, {1, 3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}}
  {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
(End)

I. Anderson, Combinatorics of Finite Sets. Oxford Univ. Press, 1987, p. 38.
Jorge Luis Arocha, "Antichains in ordered sets" [ In Spanish ]. Anales del Instituto de Matematicas de la Universidad Nacional Autonoma de Mexico 27: 1-21 (1987).
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.
J. Dezert, Fondations pour une nouvelle théorie du raisonnement plausible et paradoxal (la DSmT), Tech. Rep. 1/06769 DTIM, ONERA, Paris, page 33, January 2003.
J. Dezert, F. Smarandache, On the generating of hyper-powersets for the DSmT, Proceedings of the 6th International Conference on Information Fusion, Cairns, Australia, 2003.
M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 188.
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.
S. Muroga, Threshold Logic and Its Applications. Wiley, NY, 1971, p. 38 and 214.
D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 349.

Guillermo Alesandroni and Sungjin Kim, The probability that the Taylor resolution of a monomial ideal is minimal, J. Comm. Alg., 2024. See p. 3.
K. Atanassov, On Some of Smarandache's Problems.
K. S. Brown, Dedekind's problem.
Winfried Bruns, Pedro A. García-Sánchez, and Luca Moci, The monoid of monotone functions on a poset and arithmetic multiplicities for uniform matroids, arXiv:1902.00864 [math.CO], 2019.
Donald E. Campbell, Jack Graver, and Jerry S. Kelly, There are more strategy-proof procedures than you think, Mathematical Social Sciences 64 (2012) 263-265. - _N. J. A. Sloane_, Oct 23 2012
David Carey and Mordechai Katzman, Stanley-Reisner Ideals with Pure Resolutions, arXiv:2409.05481 [math.AC], 2024. See p. 52.
Fan Cheng, Optimality of routing on the wiretap network with simple network topology, Information Theory (ISIT), 2014 IEEE International Symposium on, June 29 2014-July 4 2014 Page(s): 786 - 790 INSPEC Accession Number: 14524545 Honolulu, HI.
Fan Cheng and Vincent Y. F. Tan, A Numerical Study on the Wiretap Network with a Simple Network Topology, arXiv preprint arXiv:1505.02862 [cs.IT], 2015.
Jean Dezert, Foundations for a new theory for plausible and paradoxical reasoning, Tech. Rep. DTIM/IED, ONERA, Paris, pp. 14-15, 2002.
Jean Dezert, Combination of paradoxical sources of information within the neutrosophic framework, Proceedings of the First International Conference on Neutrosophics (2001).
J. L. King, Brick tiling and monotone Boolean functions.
J. L. King, A change-of-coordinates from Geometry to Algebra applied to brick tilings, arXiv:math/9809176 [math.CO], 1998.
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.
Y. Nardi and A. Rinaldo, The log-linear group-lasso estimator and its asymptotic properties, Bernoulli 18(3) (2012), 945-974; see Table 2 on p. 954.
Eric Weisstein's World of Mathematics, Antichain.
R. I. P. Wickramasinghe, Topics in log-linear models, Master of Science thesis in Statistics, Texas Tech University, Lubbock, TX, 2008. [For n = 2, the a(2) = 5 hierarchical log-linear models on two factors X and Y appear on p. 18. For n = 3, the a(3) = 19 hierarchical log-linear models on three factors X, Y, and Z, appear on p. 36. - _Petros Hadjicostas_, Apr 08 2020]
D. H. Wiedemann, A computation of the eighth Dedekind number, Order 8 (1991) 5-6.
Wikipedia, Abstract simplicial complex.
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

Equals A000372 - 1 = A007153 + 1.

Cf. A003182, A005465, A006126, A006602, A058673 (labeled matroids), A058891 (labeled hypergraphs), A261005, A293606, A304996, A305000, A306505, A307249, A317674, A319721, A320449, A321679.

nn=5;
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{1,n}],SubsetQ]],{n,0,nn}] (* Gus Wiseman, Feb 20 2019 *)
A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A@372 - 1 (* Jean-François Alcover, Jan 07 2020 *)

Binomial transform of A307249 (or A006126 if its zeroth term is 1). - Gus Wiseman, Feb 20 2019

a(n) >= A005465(n) (because the hierarchical log-linear models on n factors always include all the conditional independence models considered by I. J. Good in A005465). - Petros Hadjicostas, Apr 24 2020

Last term from D. H. Wiedemann, personal communication.
Additional comments from Michael Somos, Jun 10 2002
Term a(9) (using A000372) from Joerg Arndt, Apr 07 2023

A000978 Wagstaff numbers: numbers k such that (2^k + 1)/3 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399

Offset: 1

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

It is easy to see that the definition implies that k must be an odd prime. - N. J. A. Sloane, Oct 06 2006
The terms from a(32) on only give probable primes as of 2018. Caldwell lists the largest certified primes. - Jens Kruse Andersen, Jan 10 2018
Prime numbers of the form 1+Sum_{i=1..m} 2^(2i-1). - Artur Jasinski, Feb 09 2007
There is a new conjecture stating that a Wagstaff number is prime under the following condition (based on DiGraph cycles under the LLT): Let p be a prime integer > 3, N(p) = 2^p+1 and W(p) = N(p)/3, S(0) = 3/2 (or 1/4) and S(i+1) = S(i)^2 - 2 (mod N(p)). Then W(p) is prime iff S(p-1) == S(0) (mod W(p)). - Tony Reix, Sep 03 2007
As a member of the DUR team (Diepeveen, Underwood, Reix), and thanks to the LLR tool built by Jean Penne, I've found a new and big Wagstaff PRP: (2^4031399+1)/3 is Vrba-Reix PRP! This Wagstaff number has 1,213,572 digits and today is the 3rd biggest PRP ever found. I've done a second verification on a Nehalem core with the PFGW tool. - Tony Reix, Feb 20 2010
13347311 and 13372531 were found to be terms of this sequence (maybe not the next ones) by Ryan Propper in September 2013. - Max Alekseyev, Oct 07 2013
The next term is larger than 10 million. - Gord Palameta, Mar 22 2019
Ryan Propper found another likely term, 15135397, though it only corresponds to a probable prime. - Charles R Greathouse IV, Jul 01 2021

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
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).
S. S. Wagstaff, Jr., personal communication.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
C. Caldwell's The Top Twenty, Wagstaff.
C. Caldwell, New Mersenne Conjecture
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
Editor's Note, Table of Wagstaff primes sent by D. H. Lehmer (annotated and scanned copy)
J. E. Foster, Problem 174, A solution in primes, Math. Mag., 27 (1954), 156-157.
R. K. Guy, Letter to N. J. A. Sloane, Aug 1986
R. K. Guy, Letter to N. J. A. Sloane, 1987
H. Lifchitz, Mersenne and Fermat primes field
H. & R. Lifchitz, PRP Top Records.
Henri & Renaud Lifchitz, PRP Records.
Gord Palameta, There are no new Wagstaff primes with exponent below 10 million, mersenneforum.org
Ryan Propper et al., New Wagstaff PRP exponents, mersenneforum.org
PRP top list: PRP top [From _Tony Reix_, Feb 20 2010]
Tony Reix, Yahoo Primeform Group Message 10184 dd. Feb 20, 2010, reconstruction in html.
T. Reix, Some Maths about the Vrba-Reix PRP test [From _Tony Reix_, Feb 20 2010]
Djurre G. Sikkema, Probable primality testing for Wagstaff prime, Bachelor's project mathematics, Univ. Groningen (Netherlands 2024). See p. 32.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Wagstaff Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Wikipedia, Wagstaff prime
R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

Cf. A107036 (indices of prime Jacobsthal numbers).

Cf. A000979, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A010051, A065091, A001045.

a000978 n = a000978_list !! (n-1)
a000978_list = filter ((== 1) . a010051 . a001045) a065091_list
-- Reinhard Zumkeller, Mar 24 2013

Select[Range[5000], PrimeQ[(2^# + 1)/3] &] (* Michael De Vlieger, Jan 10 2018 *)
Select[Prime[Range[2,500]],PrimeQ[(2^#+1)/3]&] (* Harvey P. Dale, Jun 13 2022 *)

forprime(p=2,5000,if(ispseudoprime(2^p\/3),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

from gmpy2 import divexact
from sympy import prime, isprime
A000978 = [p for p in (prime(n) for n in range(2,10**2)) if isprime(divexact(2**p+1,3))] # Chai Wah Wu, Sep 04 2014

a(n) = A107036(n) for n>1. - Alexander Adamchuk, Feb 10 2007

a(30) from Kamil Duszenko (kdusz(AT)wp.pl), Feb 03 2003; a(30) was proved prime by Francois Morain with FastECPP. - Tony Reix, Sep 03 2007
a(31)-a(39) from Robert G. Wilson v, Apr 11 2005
a(40) from Vincent Diepeveen (diep(AT)xs4all.nl) added by Alexander Adamchuk, Jun 19 2008
a(41) from Tony Reix, Feb 20 2010

cp's OEIS Frontend

A111441 Numbers k such that the sum of the squares of the first k primes is divisible by k.

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A128165 Numbers k such that k divides 1 plus the sum of the first k primes.

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A128166 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^2 = 1 + A024450(k).

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A000170 Number of ways of placing n nonattacking queens on an n X n board.

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A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

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A062666 Numbers k such that 100^k - 99^k is prime.

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A002982 Numbers k such that k! - 1 is prime.

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