keyword:hard - cp's OEIS frontend

A054767 Period of the sequence of Bell numbers A000110 (mod n).

1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, 25239592216021, 411771, 10153, 48, 51702516367896047761, 39, 109912203092239643840221, 9372, 1784341, 85593501183, 949112181811268728834319677753, 312, 3905, 75718776648063, 117, 1647084

Offset: 1

Eric W. Weisstein, Feb 09 2002

For p prime, a(p) divides (p^p-1)/(p-1) = A023037(p), with equality at least for p up to 19.

Wagstaff shows that N(p) = (p^p-1)/(p-1) is the period for all primes p < 102 and for primes p = 113, 163, 167 and 173. For p = 7547, N(p) is a probable prime, which means that this p may have the maximum possible period N(p) also. See A088790. - T. D. Noe, Dec 17 2008

Jianing Song, Table of n, a(n) for n = 1..102 (b-file based on the Wagstaff article)
J. Levine and R. E. Dalton, Minimum Periods, Modulo p, of First Order Bell Exponential Integrals, Mathematics of Computation, 16 (1962), 416-423.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.
Jianing Song, A summary of several proofs in the Lunnon article.
Samuel S. Wagstaff Jr., Aurifeuillian factorizations and the period of the Bell numbers modulo a prime, Math. Comp. 65 (1996), 383-391.
Eric Weisstein's World of Mathematics, Bell Number.

Cf. A000110, A023037, A214810. A146093-A146122 gives Bell numbers read mod 3 to mod 32.

(* Warning: this program is just a verification of the existing data
 and should not be used to extend the sequence beyond a(28) *)
BellMod[k_, m_] := Mod[Sum[Mod[StirlingS2[k, j], m], {j, 1, k}], m];
BellMod[k_, 1] := BellB[k];
period[nn_List] := Module[{lgmin=2, lgmax=5, nn1},
   lg=If[Length[nn]<=lgmax, lgmin, lgmax];
   nn1 = nn[[1;;lg]];
   km=Length[nn]-lg;
   Catch[Do[If[nn1==nn[[k;;k+lg-1]], Throw[k-1]];
   If[k==km, Throw[0]], {k, 2, km}]]];
dd[n_] := SelectFirst[Table[{d, n/d},
     {d, Divisors[n][[2;;-2]]}], GCD@@#==1&];
a[1]=1;
a[p_?PrimeQ] := a[p] = (p^p-1)/(p-1);
a[n_/;n>4 && dd[n]!={}] := With[{g = dd[n]}, LCM[a[g[[1]]], a[g[[2]]]]];
a[n_/;MemberQ[FactorInteger[n][[All, 2]], 1]] := a[n]=
   With[{pp = Select[FactorInteger[n], #1[[2]] ==1 &][[All, 1]]},
      a[n/Times@@pp]*Times@@a/@pp];
a[n_/;n>4 && GCD @@ FactorInteger[n][[All, 2]]>1] := a[n]=
   With[{g=GCD @@ FactorInteger[n][[All, 2]]}, n^(1/g)*a[n^(1-1/g)]];
a[n_] := period[Table[BellMod[k, n], {k, 1, 28}]];
Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Jul 31 2012, updated May 06 2024 *)

If gcd(n,m) = 1, a(n*m) = lcm(a(n), a(m)). But the sequence is not in general multiplicative; e.g. a(2) = 3, a(9) = 39 and a(18) = 39. - Franklin T. Adams-Watters, Jun 06 2006

a(2^s) = 3*2^s for s >= 2 (Theorem 6.4 in the Lunnon article). For an odd prime p, if a(p) = (p^p-1)/(p-1) (which is conjectured to hold for all p), then a(p^s) = p^(s-1)*(p^p-1)/(p-1) (Theorem 6.2 in the Lunnon article). - Jianing Song, Jun 18 2025

More information from Phil Carmody, Dec 22 2002
Extended by T. D. Noe, Dec 18 2008
a(26) corrected by Jean-François Alcover, Jul 31 2012
a(18) corrected by Charles R Greathouse IV, Jul 31 2012
a(27)-a(28) from Charles R Greathouse IV, Sep 07 2016

A004061 Numbers k such that (5^k - 1)/4 is prime.

Original entry on oeis.org

3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, 3300593

Offset: 1

N. J. A. Sloane

With the addition of the 19th prime in the sequence, the new best linear fit to the sequence has G=0.4723, which is slightly closer to the conjectured limit of G=0.56145948, A080130 (see link for Generalized Repunit Conjecture). - Paul Bourdelais, Apr 30 2018

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paul Bourdelais, A Generalized Repunit Conjecture [From _Paul Bourdelais_, Jun 01 2010]
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Index to primes in various ranges, form ((k+1)^n-1)/k

Cf. A080130.

lst={};Do[If[PrimeQ[(5^n-1)/4], AppendTo[lst, n]], {n, 10^4}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 20 2008 *)

forprime(p=2,1e4,if(ispseudoprime(5^p\4),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(13)-a(15) from Kamil Duszenko (kdusz(AT)wp.pl), Mar 25 2003
a(16) corresponds to a probable prime based on trial factoring to 4*10^13 and Fermat primality testing base 2. - Paul Bourdelais, Dec 11 2008
a(17) corresponds to a probable prime discovered by Paul Bourdelais, Jun 01 2010
a(18) corresponds to a probable prime discovered by Paul Bourdelais, Apr 30 2018
a(19) corresponds to a probable prime discovered by Ryan Propper, Jan 02 2022

A054377 Primary pseudoperfect numbers: numbers n > 1 such that 1/n + sum 1/p = 1, where the sum is over the primes p | n.

Original entry on oeis.org

2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086

Offset: 1

Eric W. Weisstein

Primary pseudoperfect numbers are the solutions of the "differential equation" n' = n-1, where n' is the arithmetic derivative of n. - Paolo P. Lava, Nov 16 2009
Same as n > 1 such that 1 + sum n/p = n (and the only known numbers n > 1 satisfying the weaker condition that 1 + sum n/p is divisible by n). Hence a(n) is squarefree, and is pseudoperfect if n > 1. Remarkably, a(n) has exactly n (distinct) prime factors for n < 9. - Jonathan Sondow, Apr 21 2013
From the Wikipedia article: it is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers. - Daniel Forgues, May 27 2013
Since the arithmetic derivative of a prime p is p' = 1, 2 is obviously the only prime in the sequence. - Daniel Forgues, May 29 2013
Just as 1 is not a prime number, 1 is also not a primary pseudoperfect number, according to the original definition by Butske, Jaje, and Mayernik, as well as Wikipedia and MathWorld. - Jonathan Sondow, Dec 01 2013
Is it always true that if a primary pseudoperfect number N > 2 is adjacent to a prime N-1 or N+1, then in fact N lies between twin primes N-1, N+1? See A235139. - Jonathan Sondow, Jan 05 2014
Also, integers n > 1 such that A069359(n) = n - 1. - Jonathan Sondow, Apr 16 2014

			From _Daniel Forgues_, May 24 2013: (Start)
With a(1) = 2, we have 1/2 + 1/2 = (1 + 1)/2 = 1;
with a(2) = 6 = 2 * 3, we have
  1/2 + 1/3 + 1/6 = (3 + 2 + 1)/6 = (1*3 + 3)/(2*3) = (1 + 1)/2 = 1;
with a(3) = 42 = 6 * 7, we have
  1/2 + 1/3 + 1/7 + 1/42 = (21 + 14 + 6 + 1)/42 =
  (3*7 + 2*7 + 7)/(6*7) = (3 + 2 + 1)/6 = 1;
with a(4) = 1806 = 42 * 43, we have
  1/2 + 1/3 + 1/7 + 1/43 + 1/1806 = (903 + 602 + 258 + 42 + 1)/1806 =
  (21*43 + 14*43 + 6*43 + 43)/(42*43) = (21 + 14 + 6 + 1)/42 = 1;
with a(5) = 47058 (not oblong number), we have
  1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/47058 =
  (23529 + 15686 + 4278 + 2046 + 1518 + 1)/47058 = 1.
For n = 1 to 8, a(n) has n prime factors:
  a(1) = 2
  a(2) = 2 * 3
  a(3) = 2 * 3 *  7
  a(4) = 2 * 3 *  7 * 43
  a(5) = 2 * 3 * 11 * 23 *  31
  a(6) = 2 * 3 * 11 * 23 *  31 * 47059
  a(7) = 2 * 3 * 11 * 17 * 101 *   149 *       3109
  a(8) = 2 * 3 * 11 * 23 *  31 * 47059 * 2217342227 * 1729101023519
If a(n)+1 is prime, then a(n)*[a(n)+1] is also primary pseudoperfect. We have the chains: a(1) -> a(2) -> a(3) -> a(4); a(5) -> a(6). (End)
A primary pseudoperfect number (greater than 2) is oblong if and only if it is not the initial member of a chain. - _Daniel Forgues_, May 29 2013
If a(n)-1 is prime, then a(n)*(a(n)-1) is a Giuga number (A007850). This occurs for a(2), a(3), and a(5). See A235139 and the link "The p-adic order . . .", Theorem 8 and Example 1. - _Jonathan Sondow_, Jan 06 2014

Max Alekseyev, Jose Maria Grau, and Antonio Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT], 2016-2018.
Thomas Bloom, Are there infinitely many solutions to 1/p_1 + ... + 1/p_k = 1 - 1/m, where m >= 2 is an integer and p_1 < ... < p_k are distinct primes?, Erdős Problems.
William Butske, Lynda M. Jaje and Daniel R. Mayernik, On the Equation Sum_{p|N} 1/p + 1/N = 1, Pseudoperfect numbers and partially weighted graphs, Math. Comput., 69 (1999), 407-420. [Title corrected by _Jonathan Sondow_, Apr 11 2012]
José María Grau, Antonio M. Oller-Marcén, and Jonathan Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv:1309.7941 [math.NT], 2013.
José María Grau, Antonio M. Oller-Marcén, and Daniel Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.
Jonathan Sondow and Kieren MacMillan, Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2, Integers 11 (2011), #A34.
Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.
Jonathan Sondow and Emmanuel Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Terence Tao, Erdős problem database, see no. 313.
Eric Weisstein's World of Mathematics, Primary pseudoperfect number.
Wikipedia, Primary pseudoperfect number.
OEIS Wiki, Primary pseudoperfect numbers.

Cf. A005835, A007850, A069359, A168036, A190272, A191975, A203618, A216825, A216826, A230311, A235137, A235138, A235139, A236433.

pQ[n_] := (f = FactorInteger[n]; 1/n + Sum[1/f[[i]][[1]], {i, Length[f]}] == 1)
Select[Range[2, 10^6], pQ[#] &] (* Robert Price, Mar 14 2020 *)

isok(n) = if (n > 1, my(f=factor(n)[,1]); 1/n + sum(k=1, #f, 1/f[k]) == 1); \\ Michel Marcus, Oct 05 2017

from sympy import primefactors
A054377 = [n for n in range(2,10**5) if sum([n/p for p in primefactors(n)]) +1 == n] # Chai Wah Wu, Aug 20 2014

A031971(a(n)) (mod a(n)) = A233045(n). - Jonathan Sondow, Dec 11 2013
A069359(a(n)) = a(n) - 1. - Jonathan Sondow, Apr 16 2014
a(n) == 36*(n-2) + 6 (mod 288) for n = 2,3,..,8. - Kieren MacMillan and Jonathan Sondow, Sep 20 2017

A000104 Number of n-celled free polyominoes without holes.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 35, 107, 363, 1248, 4460, 16094, 58937, 217117, 805475, 3001127, 11230003, 42161529, 158781106, 599563893, 2269506062, 8609442688, 32725637373, 124621833354, 475368834568, 1816103345752, 6948228104703, 26618671505989, 102102788362303

Offset: 0

N. J. A. Sloane

J. S. Madachy, Pentominoes - Some Solved and Unsolved Problems, J. Rec. Math., 2 (1969), 181-188.
George E. Martin, Polyominoes - A Guide to Puzzles and Problems in Tiling, The Mathematical Association of America, 1996
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).

John Mason, Table of n, a(n) for n = 0..40
Elena V. Konstantinova and Maxim V. Vidyuk, Discriminating tests of information and topological indices. Animals and trees, J. Chem. Inf. Comput. Sci. 43 (2003), 1860-1871.
John Mason, Description of counting programs
John Mason, Programs for calculation of numbers of unholey polyominoes
Lucia Moura and Ivan Stojmenovic, Backtracking and Isomorph-Free Generation of Polyhexes, Table 2.2 on p. 55 of Handbook of Applied Algorithms (2008).
W. R. Muller, K. Szymanski, J. V. Knop, and N. Trinajstic, On the number of square-cell configurations, Theor. Chim. Acta 86 (1993) 269-278
Tomás Oliveira e Silva, Enumeration of polyominoes
T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy)
R. C. Read, Contributions to the cell growth problem, Canad. J. Math., 14 (1962), 1-20.

Cf. A000105, row sums of A308300, A006746, A056877, A006748, A056878, A006747, A006749, A054361, A070765 (polyiamonds), A018190 (polyhexes), A266549 (by perimeter).

Cf. A056879, A056880, A056881, A056882, A006724, A357647, A357648.

a(n) = A000105(n) - A001419(n). - John Mason, Sep 06 2022

a(n) = (4*A056879(n) + 4*A056881(n) + 4*A056883(n) + 6*A056880(n) + 6*A056882(n) + 6*A357647(n) + 7*A357648(n) + A006724(n)) / 8. - John Mason, Oct 10 2022

Extended to n=26 by Tomás Oliveira e Silva

a(27)-a(28) from Tomás Oliveira e Silva's page added by Andrey Zabolotskiy, Oct 02 2022

A006506 Number of n X n binary matrices with no 2 adjacent 1's, or number of configurations of non-attacking princes on an n X n board, where a "prince" attacks the four adjacent (non-diagonal) squares. Also number of independent vertex sets in an n X n grid.

Original entry on oeis.org

1, 2, 7, 63, 1234, 55447, 5598861, 1280128950, 660647962955, 770548397261707, 2030049051145980050, 12083401651433651945979, 162481813349792588536582997, 4935961285224791538367780371090, 338752110195939290445247645371206783, 52521741712869136440040654451875316861275

Offset: 0

N. J. A. Sloane, R. H. Hardin, Paul Zimmermann

A two-dimensional generalization of the Fibonacci numbers.
Also the number of vertex covers in the n X n grid graph P_n X P_n.
A181030 (Number of n X n binary matrices with no leading bitstring in any row or column divisible by 4) is the same sequence. Proof from Steve Butler, Jan 26 2015: This is trivially true. A181030 is equivalent to this sequence by interchanging the roles of 0 and 1. In particular, A181030 looks for binary matrices with no leading bitstring divisible by 4, but a bitstring is divisible by 4 if and only if its last two digits is 0; in a binary matrix this can only be avoided if there are no two adjacent 0's (i.e., for any two adjacent 0's take the bitstring starting in that row or column and we are done); the present sequence looks for no two adjacent 1's. Similar reasons show that the array A181031 is equivalent to the array A089980.
Let R(n) be the set of squares that have vertices at integer coordinates and lie in the region of the plane |x|+|y| <= n+1, and let S(n) be the set of squares that have vertices at integer coordinates and lie in the region of the plane |x|+|y-1/2| <= n+2.  Further let T be the collection of rectangular tiles with dimensions i X 1 or 1 X i with i arbitrary.  Then a(2n) is the number of ways to tile R(n) using tiles from T and a(2n+1) is the number of ways to tile S(n) using tiles from T. (Note R(n) is the Aztec diamond of order n.) - Steve Butler, Jan 26 2015

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 342-349.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jin-Guo Liu, Table of n, a(n) for n = 0..39 (terms 1..33 from Robert Gerbicz, 34..35 from P. Butera and M. Pernici, 37..38 from Casey Mills Davis)
P. Butera and M. Pernici, Sums of permanental minors using Grassmann algebra, arXiv preprint arXiv:1406.5337 [hep-lat], 2014.
Casey Mills Davis, C++ program used to generate a(n) for n = 36..37
Steven R. Finch, Hard Square Entropy Constant [Broken link]
Steven R. Finch, Hard Square Entropy Constant [From the Wayback machine]
Xuanzhao Gao, Xiaofeng Li, and Jinguo Liu, Programming guide for solving constraint satisfaction problems with tensor networks, arXiv:2501.00227 [physics.comp-ph], 2024. See p. 24.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.372.
Jin-Guo Liu, Xun Gao, Madelyn Cain, Mikhail D. Lukin and Sheng-Tao Wang, Computing solution space properties of combinatorial optimization problems via generic tensor networks, arXiv:2205.03718 [cond-mat.stat-mech], 2022. Terms 38..39.
I. Mezo, Periodicity of the Last Digits of Some Combinatorial Sequences, J. Int. Seq. 17 (2014) #14.1.1.
B. D. Stosic, T. Stosic, I. P. Fittipaldi and J. J. P. Veerman, Residual entropy of the square Ising antiferromagnet in the maximum critical field: the Fibonacci matrix, Journal of Physics A: Mathematical and General, Volume 30, Number 10, 1997, pp. L331-L337.
Peter Tittmann, Enumeration in Graphs
Eric Weisstein's World of Mathematics, (0,1)-Matrix
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Hard Square Entropy Constant
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for sequences related to binary matrices

Cf. A027683 for toroidal version.
Table of values for n x m matrices: A089934.
Cf. A232833 for refinement by number of 1's.
Cf. also A191779.
Cf. A201511, A212270.

A006506 := proc(N) local i,j,p,q; p := 1+x11;
if n=0 then return 1 fi;
for i from 2 to N do
   q := p-select(has,p,x||(i-1)||1);
   p := p+expand(q*x||i||1)
od;
for j from 2 to N do
   q := p-select(has,p,x1||(j-1));
   p := subs(x1||(j-1)=1,p)+expand(q*x1||j);
   for i from 2 to N do
      q := p-select(has,p,{x||(i-1)||j,x||i||(j-1)});
      p := subs(x||i||(j-1)=1,p)+expand(q*x||i||j);
   od
od;
map(icontent,p)
end:
seq(A006506(n), n=0..15);

a[n_] := a[n] = (p = 1 + x[1, 1]; Do[q = p - Select[p, ! FreeQ[#, x[i-1, 1]] &]; p = p + Expand[q*x[i, 1]], {i, 2, n}]; Do[q = p - Select[p, ! FreeQ[#, x[1, j-1]] &]; p = (p /. x[i, j-1] :> 1) + Expand[q*x[1, j]]; Do[q = p - Select[ p, ! FreeQ[#, x[i-1, j]] || ! FreeQ[#, x[i, j-1]] &]; p = (p /. x[i, j-1] :> 1) + Expand[q*x[i, j]], {i, 2, n}], {j, 2, n}]; p /. x[, ] -> 1); a /@ Range[14] (* Jean-François Alcover, May 25 2011, after Maple prog. *)
Table[With[{g = GridGraph[{n, n}]}, Count[Subsets[Range[n^2], Length @ First @ FindIndependentVertexSet[g]], ?(IndependentVertexSetQ[g, #] &)]], {n, 5}] (* _Eric W. Weisstein, May 28 2017 *)

a(n)=L=fibonacci(n+2);p=v=vector(L,i,1);c=0; for(i=0,2^n-1,j=i;while(j&&j%4<3,j\=2);if(j%4<3,p[c++]=i)); for(i=2,n,w=vector(L,j,0); for(j=1,L, for(k=1,L,if(bitand(p[j],p[k])==0,w[j]+=v[k])));v=w); sum(i=1,L,v[i]) \\ Robert Gerbicz, Jun 17 2011

Limit_{n->oo} a(n)^(1/n^2) = c1 = 1.50304... is the hard square entropy constant A085850. - Benoit Cloitre, Nov 16 2003
a(n) appears to behave like A * c3^n * c1^(n^2) where c1 is as above, c3 = 1.143519129587 approximately, A = 1.0660826 approximately. This is based on numerical analysis of a(n) for n up to 19. - Brendan McKay, Nov 16 2003
From n up to 39 we have A = 1.06608266035... - Vaclav Kotesovec, Jan 29 2024

Sequence extended by Paul Zimmermann, Mar 15 1996
Maple program updated and sequence extended by Robert Israel, Jun 16 2011
a(0)=1 prepended by Alois P. Heinz, Jan 29 2024

A127816 a(n) = least k >= 1 such that the remainder when 6^k is divided by k is n.

Original entry on oeis.org

5, 34, 213, 68, 4021227877, 7, 121129, 14, 69, 26, 767, 51, 6191, 22, 201, 20, 1919, 33, 169, 44, 39, 1778, 1926049, 174, 2673413, 50, 63, 451, 1257243481237, 93, 851, 316, 183, 14809, 1969, 38, 1362959, 1826, 177, 289, 65, 87, 5567, 1252, 57, 1651, 6403249

Offset: 1

Alexander Adamchuk, Jan 30 2007, Feb 05 2007

a(7^k-1) = 7^k.

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

Cf. A036236, A078457, A119678, A119679, A119715, A119714, A127817, A127818, A127819, A127820, A127821.

t = Table[0, {10000}]; k = 1; lst = {}; While[k < 5600000000, a = PowerMod[6, k, k]; If[ a<10001 && t[[a]]==0, t[[a]]=k; Print[{a,k}]]; k++ ]; t

a(7^k-1) = 7^k.

a(5) from Joe K. Crump confirmed and a(6)-a(28) added by Ryan Propper, Feb 21 2007
I combined the two Mathematica codings into one and extended the search limits. - Robert G. Wilson v, Jul 16 2009
a(29) as conjectured by J. K. Crump confirmed by Hagen von Eitzen, Jul 21 2009
Corrected authorship of the a-file - R. J. Mathar, Aug 24 2009

A127818 a(n) is the least k such that the remainder when 10^k is divided by k is n.

Original entry on oeis.org

3, 14, 7, 6, 35, 94, 993, 46, 22963573117, 11, 15084115509707, 22, 21, 86, 99985, 24, 221819, 82, 327, 1996, 28039, 26, 169, 38, 39, 74, 24257, 36, 10191082613, 65, 49, 34, 4739, 66, 99965, 188, 171, 62, 3753219157, 60, 3961, 58, 87, 76, 28315, 159, 10441

Offset: 1

Alexander Adamchuk, Jan 30 2007

a(9) <= 22963573117 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 09 2007
a(11) <= 15084115509707 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 06 2007
a(29) <= 112237795073 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 09 2007
a(39) <= 3753219157 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 10 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, A127819, A127820, A127821.

t = Table[0, {10000} ]; k = 1; While[ k < 4000000000, a = PowerMod[10, k, k]; If[a < 10001 && t[[a]] == 0, t[[a]] = k; Print[{a, k}]]; k++ ]; t

def a(n):
  k = 1
  while 10**k % k != n: k += 1
  return k
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, Mar 14 2021

Crump's values for a(9), a(11), a(39) confirmed, a(29) = 10191082613 = 16763 * 607951 by Hagen von Eitzen, Jul 29 2009

A005470 Number of unlabeled planar simple graphs with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 11, 33, 142, 822, 6966, 79853, 1140916, 18681008, 333312451

Offset: 0

N. J. A. Sloane

Euler transform of A003094. - Christian G. Bower

			a(2) = 2 since o o and o-o are the two planar simple graphs on two nodes.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Trotter, ed., Planar Graphs, Vol. 9, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Amer. Math. Soc., 1993.
Turner, James; Kautz, William H. A survey of progress in graph theory in the Soviet Union. SIAM Rev. 12 1970 suppl. iv+68 pp. MR0268074 (42 #2973). See p. 19. - N. J. A. Sloane, Apr 08 2014
Vetukhnovskii, F. Ya. "Estimate of the Number of Planar Graphs." In Soviet Physics Doklady, vol. 7, pp. 7-9. 1962. - From N. J. A. Sloane, Apr 08 2014
R. J. Wilson, Introduction to Graph Theory. Academic Press, NY, 1972, p. 162.

G. Brinkmann, and B. D. McKay, Fast generation of planar graphs, MATCH Commun. Math. Comput. Chem., 58 (2007) 323-357.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph.
Gunnar Brinkmann and Brendan McKay, plantri and fullgen programs for generation of certain types of planar graph [Cached copy, pdf file only, no active links, with permission]
E. Friedman, Illustration of small graphs
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Planar Graph
Index entries for "core" sequences

Cf. A003094 (connected planar graphs), A034889, A039735 (planar graphs by nodes and edges).

Cf. A126201.

A003094 = Cases[Import["https://oeis.org/A003094/b003094.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A003094] (* Jean-François Alcover, Apr 25 2013, updated Mar 17 2020 *)

n=8 term corrected and n=9..11 terms calculated by Brendan McKay
Terms a(0) - a(10) confirmed by David Applegate and N. J. A. Sloane, Mar 09 2007
a(12) added by Vaclav Kotesovec after A003094 (computed by Brendan McKay), Dec 06 2014

A007540 Wilson primes: primes p such that (p-1)! == -1 (mod p^2).

Original entry on oeis.org

5, 13, 563

Offset: 1

N. J. A. Sloane

Suggested by the Wilson-Lagrange Theorem: An integer p > 1 is a prime if and only if (p-1)! == -1 (mod p). Cf. Wilson quotients, A007619.
Sequence is believed to be infinite. Next term is known to be > 2*10^13 (cf. Costa et al., 2013).
Intersection of the Wilson numbers A157250 and the primes A000040. - Jonathan Sondow, Mar 04 2016
Conjecture: Odd primes p such that 1^(p-1) + 2^(p-1) + ... + (p-1)^(p-1) == p-1 (mod p^2). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
From Felix Fröhlich, Nov 16 2018: (Start)
Harry S. Vandiver apparently said about the Wilson primes "It is not known if there are infinitely many Wilson primes. This question seems to be of such a character that if I should come to life any time after my death and some mathematician were to tell me that it had definitely been settled, I think I would immediately drop dead again." (cf. Ribenboim, 2000, p. 217).
Let p be a Wilson prime and let i be the index of p in A000040. For n = 1, 2, 3, the values of i are 3, 6, 103. The primes among those values are Lerch primes, i.e., terms of A197632. Is this a property that necessarily follows if i is prime (cf. Sondow, 2011/2012, 2.5 Open Problems 5)? (End)
From Amiram Eldar, Jun 16 2021: (Start)
Named after the English mathematician John Wilson (1741-1793) after whom "Wilson's theorem" was also named.
The primes 5 and 13 appear in an exercise involving the Wilson congruence in Mathews (1892). [Edited by Felix Fröhlich, Jul 23 2021]
Beeger found that there are no other smaller terms up to 114 (1913) and up to 200 (1930).
a(3) = 563 was found by Goldberg (1953), who used the Bureau of Standards Eastern Automatic Computer (SEAC) to search all primes less than 10000. According to Goldberg, the third prime was discovered independently by Donald Wall six month later. (End)

N. G. W. H. Beeger, On the Congruence (p-1)! == -1 (mod p^2), Messenger of Mathematics, Vol. 49 (1920), pp. 177-178.
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 52.
Calvin C. Clawson, Mathematical Mysteries, Plenum Press, 1996, p. 180.
Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 29.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 80.
G. B. Mathews, Theory of Numbers Part I., Cambridge: Deighton, Bell and Co., London: George Bell and Sons, 1892, page 318.
Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer Science & Business Media, 2000, ISBN 0-387-98911-0.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 277.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 234-235.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ilan Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 73.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 163.

N. G. W. H. Beeger, Quelques remarques sur les congruences r^(p-1) == 1 (mod p^2) et (p- 1)! == -1 (mod p^2), The Messenger of Mathematics, Vol. 43 (1913), pp. 72-84.
Edgar Costa, Robert Gerbicz and David Harvey, A search for Wilson primes, Mathematics of Computation, Vol. 83, No. 290 (2014), pp. 3071-3091; arXiv preprint, arXiv:1209.3436 [math.NT], 2012.
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Mathematics of Computation, 66 (1997), 433-449.
Karl Goldberg, A Table of Wilson Quotients and the Third Wilson Prime, Journal of the London Mathematical Society, Vol. 28 (1953), pp. 252-256.
James Grime and Brady Haran, What do 5, 13 and 563 have in common?, YouTube video (2014).
Emma Lehmer, A Note on Wilson's Quotient, The American Mathematical Monthly, Vol. 44, No. 4 (1937), pp. 237-238.
Emma Lehmer, On the Congruence (p-1)! == -1 (mod p^2), The American Mathematical Monthly, Vol. 44, No. 7 (1937), p. 462.
Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson", Annals of Mathematics, Vol. 39, No. 2 (1938), pp. 350-360.
George Ballard Mathews, Theory of numbers, Part I, Cambridge, 1892, p. 318.
Tapio Rajala, Status of a search for Wilson primes
Maxie D. Schmidt, New Congruences and Finite Difference Equations for Generalized Factorial Functions, arXiv:1701.04741 [math.CO], 2017.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-Non-Wilson Primes 2, 3, 14771, In: M. B. Nathanson, Combinatorial and Additive Number Theory, Springer, CANT 2011 and 2012. Also on arXiv, arXiv:1110.3113 [math.NT], 2011-2012.
Apoloniusz Tyszka, On sets X subset of N for which we know an algorithm that computes a threshold number t(X) in N such that X is infinite if and only if X contains an element greater than t(X), 2019.
Eric Weisstein's World of Mathematics, Wilson Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Wilson prime.
Paul Zimmermann, Records for prime numbers.

Cf. A007619, A157249, A157250, A377266.

Select[Prime[Range[500]], Mod[(# - 1)!, #^2] == #^2 - 1 &] (* Harvey P. Dale, Mar 30 2012 *)

forprime(n=2, 10^9, if(Mod((n-1)!, n^2)==-1, print1(n, ", "))) \\ Felix Fröhlich, Apr 28 2014

is(n)=prod(k=2,n-1,k,Mod(1,n^2))==-1 \\ Charles R Greathouse IV, Aug 03 2014

from sympy import prime
A007540_list = []
for n in range(1,10**4):
    p, m = prime(n), 1
    p2 = p*p
    for i in range(2,p):
        m = (m*i) % p2
    if m == p2-1:
        A007540_list.append(p) # Chai Wah Wu, Dec 04 2014

A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444

Offset: 0

Robert G. Wilson v, Sep 06 2000

Smallest base value yielding generalized Fermat primes. - Hugo Pfoertner, Jul 01 2003
The first 5 terms correspond with the known (ordinary) Fermat primes. A probable candidate for the next entry is 62722^131072+1, discovered by Michael Angel in 2003. It has 628808 decimal digits. - Hugo Pfoertner, Jul 01 2003
For any n, a(n+1) >= sqrt(a(n)), because k^(2^(n+1))+1 = (k^2)^(2^n)+1. - Jeppe Stig Nielsen, Sep 16 2015
Does the sequence contain any perfect squares? If a(n) is a perfect square, then a(n+1) = sqrt(a(n)). - Jeppe Stig Nielsen, Sep 16 2015
If for a particular n, a(n) exists, then a(i) exist for all i=0,1,2,...,n. No proof is known that this sequence is infinite. Such a result would clearly imply the infinitude of A002496. - Jeppe Stig Nielsen, Sep 18 2015
919444 is a candidate for a(20). See Zimmermann link. -  Serge Batalov, Sep 02 2017
Now PrimeGrid has tested and double checked all b^(2^20) + 1 with b < 919444, so we have proof that a(20) = 919444. - Jeppe Stig Nielsen, Dec 30 2017

			The primes are 2^(2^0) + 1 = 3, 2^(2^1) + 1 = 5, 2^(2^2) + 1 = 17, 2^(2^3) + 1 = 257, 2^(2^4) + 1 = 65537, 30^(2^5) + 1, 102^(2^6) + 1, ....

Yves Gallot, Generalized Fermat Prime Search
Lucile and Yves Gallot, Generalized Fermat Prime Search
Michael Goetz, id=103235 of Top 5000 Primes
Luke Harmon, Gaetan Delavignette, Arnab Roy, and David Silva, PIE: p-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption, Cryptology ePrint Archive 2023/700.
Stephen Scott, id=84401 of Top 5000 Primes
Sylvanus A. Zimmerman, PrimeGrid’s Generalized Fermat Prime Search

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

Cf. A019434 (Fermat primes).

f[n_] := (p = 2^n; k = 2; While[cp = k^p + 1; !PrimeQ@cp, k++ ]; k); Do[ Print[{n, f@n}], {n, 0, 17}] (* Lei Zhou, Feb 21 2005 *)

a(n)=my(k=2);while(!isprime(k^(2^n)+1),k++);k \\ Anders Hellström, Sep 16 2015

a(n) = A085398(2^(n+1)). - Jianing Song, Jun 13 2022

from Robert G. Wilson v, Oct 30 2000
from Jeppe Stig Nielsen, Aug 07 2005
and 75898 from Lei Zhou, Feb 01 2012
919444 from Jeppe Stig Nielsen, Dec 30 2017

cp's OEIS Frontend

A054767 Period of the sequence of Bell numbers A000110 (mod n).

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A004061 Numbers k such that (5^k - 1)/4 is prime.

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A054377 Primary pseudoperfect numbers: numbers n > 1 such that 1/n + sum 1/p = 1, where the sum is over the primes p | n.

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A000104 Number of n-celled free polyominoes without holes.

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A006506 Number of n X n binary matrices with no 2 adjacent 1's, or number of configurations of non-attacking princes on an n X n board, where a "prince" attacks the four adjacent (non-diagonal) squares. Also number of independent vertex sets in an n X n grid.

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A127816 a(n) = least k >= 1 such that the remainder when 6^k is divided by k is n.

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A127818 a(n) is the least k such that the remainder when 10^k is divided by k is n.

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