keyword:hard - cp's OEIS frontend

A127817 a(n) = least k such that the remainder when 9^k is divided by k is n.

2, 7, 6, 5, 38, 723, 74, 2592842671511, 11, 3827, 14, 717, 34, 59035, 21, 259, 152, 237, 62, 626131, 30, 169, 58, 25, 56, 1921, 39, 361, 65, 49, 63010, 287, 48, 55, 46, 63, 932, 3786791, 69, 69637, 230, 221, 6707, 1057, 57, 4907, 253, 681, 148, 393217991, 70

Offset: 1

Alexander Adamchuk, Jan 30 2007

			For n=4, since 9^5 == 4 (mod 5) and 9^k is not congruent to 4 (mod k) for any k < 5, a(4) = 5. _Michael B. Porter_, Dec 10 2016

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..10000 with -1 for large entries where a(n) has not yet been found, Nov 21 2016 [With 202 new terms, this supersedes an earlier table of Robert G. Wilson v et al.]

Cf. A036236, A078457, A119678, A119679, A127816, A119715, A119714, A127818, A127819, A127820, A127821.

a127817 := [seq(0,j=1..nmax)] ; for k from 1 do n := modp(9^k,k) ; if n > 0 and n <= nmax then if op(n,a127817) = 0 then a127817 := subsop(n=k,a127817) ; print( op(1..50,a127817) ) ; fi; fi; od: # R. J. Mathar, Jul 16 2009

t = Table[0, {10000}]; k = 1; lst = {}; While[k < 4500000000, a = PowerMod[9, k, k]; If[ a<10001 && t[[a]]==0, t[[a]]=k; Print[{a,k}]]; k++ ]; t

a(8) <= 2592842671511 from Joe K. Crump (joecr(AT)carolina.rr.com), Feb 06 2007
I changed the Mathematica coding to reflect the current limits Robert G. Wilson v, Jul 18 2009
Value for a(8) as suggested by J. K. Crump confirmed by Hagen von Eitzen, Jul 21 2009
Authorship of a-file corrected by R. J. Mathar, Aug 24 2009

A229542 Numbers n such that (19^n - 2^n)/17 is prime.

Original entry on oeis.org

11, 19, 79, 631, 1787, 2011, 2381, 20219, 49523

Offset: 1

Robert Price, Sep 25 2013

All terms are primes.

a(10) > 2*10^5.

Cf. A128032, A225955, A225807, A006035.

Select[ Prime[ Range[1, 100000] ], PrimeQ[ (19^# - 2^#)/17 ]& ]

is(n)=ispseudoprime((19^n-2^n)/17) \\ Charles R Greathouse IV, Jun 13 2017

A005176 Number of regular graphs with n unlabeled nodes.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 8, 6, 22, 26, 176, 546, 19002, 389454, 50314870, 2942198546, 1698517037030, 442786966117636, 649978211591622812, 429712868499646587714, 2886054228478618215888598, 8835589045148342277802657274, 152929279364927228928025482936226, 1207932509391069805495173417972533120, 99162609848561525198669168653641835566774

Offset: 0

N. J. A. Sloane

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

E. Friedman, Illustration of small graphs
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Jennifer M. Larson, Cheating Because They Can: Social Networks and Norm Violators, 2014. See Footnote 11.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Regular Graph.

Not necessarily connected simple regular graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Simple regular graphs of any degree: A005177 (connected), A068932 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs with girth at least g: this sequence (g=3), A185314 (g=4), A185315 (g=5), A185316 (g=6), A185317 (g=7), A185318 (g=8), A185319 (g=9).
Cf. A295193.

a(n) = A005177(n) + A068932(n). - David Wasserman, Mar 08 2002

Row sums of triangle A051031.

More terms from David Wasserman, Mar 08 2002
a(15) and a(16) from Jason Kimberley, Sep 25 2009
Edited by Jason Kimberley, Jan 06 2011 and May 24 2012
a(17)-a(21) from Andrew Howroyd, Mar 08 2020
a(22)-a(24) from Andrew Howroyd, Apr 05 2020

A014127 Mirimanoff primes: primes p such that p^2 divides 3^(p-1) - 1.

Original entry on oeis.org

11, 1006003

Offset: 1

N. J. A. Sloane

Dorais and Klyve proved that there are no further terms up to 9.7*10^14.
These primes are so named after the celebrated result of Mirimanoff in 1910 (see below) that for a failure of the first case of Fermat's Last Theorem, the exponent p must satisfy the criterion stated in the definition. Lerch (see below) showed that these primes also divide the numerator of the harmonic number H(floor(p/3)). This is analogous to the fact that Wieferich primes (A001220) divide the numerator of the harmonic number H((p-1)/2). - John Blythe Dobson, Mar 02 2014, Apr 09 2015
The prime 1006003 was apparently discovered by K. E. Kloss (cf. Kloss, 1965) according to various sources. - Felix Fröhlich, Dec 08 2020
If there is no term other than 11 and 1006003, then the only solution (a, w, x, y, z) to the diophantine equation a^w + a^x = 3^y + 3^z is (5, 1, 1, 2, 3) (cf. Scott, Styer, 2006, Lemma 12). - Felix Fröhlich, Dec 10 2020
Named after the Russian mathematician Dmitry Semionovitch Mirimanoff (1861-1945). - Amiram Eldar, Jun 10 2021

Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem, Springer, 1979, pp. 23, 152-153.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 233.
Alf van der Poorten, Notes on Fermat's Last Theorem, Wiley, 1996, p. 21.

Amir Akbary and Sahar Siavashi, The Largest Known Wieferich Numbers, INTEGERS, 18(2018), A3. See Table 1 p. 5.
Chris K. Caldwell, Fermat Quotient, The Prime Glossary.
John Blythe Dobson, On the special harmonic numbers H_floor(p/9) and H_floor(p/18) modulo p, arXiv:2302.02027 [math.NT], 2023.
François G. Dorais and Dominic Klyve, A Wieferich prime search up to p < 6.7*10^15, J. Integer Seq., Vol. 14 (2011), Article 11.9.2, 1-14.
Wilfrid Keller and Jörg Richstein, Solutions of the congruence a^(p-1) == 1 (mod p^r), Math. Comp., Vol. 74, No. 250 (2005), pp. 927-936.
K. E. Kloss, Some Number-Theoretic Calculations, Journal of Research of the National Bureau of Standards - B. Mathematics and Mathematical Physics, Vol. 69B, No. 4 (Oct.-Dec. 1965), pp. 335-336.
Mathias Lerch, Zur Theorie des Fermatschen Quotienten (a^(p-1) - 1)/p == q(a), Mathematische Annalen, Vol. 60 (1905), pp. 471-490.
D. Mirimanoff, Sur le dernier théorème de Fermat, C. R. Acad. Sci. Paris, Vol. 150 (1910), pp. 204-206. Revised as Sur le dernier théorème de Fermat, Journal für die reine und angewandte Mathematik, Vol. 139 (1911), pp. 309-324.
Planet Math, Wieferich Primes.
Reese Scott and Robert Styer, On the generalized Pillai equation +-a^x +-b^y = c, Journal of Number Theory, Vol. 118, No. 2 (2006), pp. 236-265.

Cf. A039951, A096082.

Sequences "primes p such that p^2 divides X^(p-1)-1": A001220 (X=2), A123692 (X=5), A212583 (X=6), A123693 (X=7), A045616 (X=10), A111027 (X=12), A128667 (X=13), A234810 (X=14), A242741 (X=15), A128668 (X=17), A244260 (X=18), A090968 (X=19), A242982 (X=20), A298951 (X=22), A128669 (X=23), A306255 (X=26), A306256 (X=30).

Select[Prime[Range[1000000]], PowerMod[3, # - 1, #^2] == 1 &] (* Robert Price, May 17 2019 *)

N=10^9; default(primelimit,N);
forprime(n=2,N,if(Mod(3,n^2)^(n-1)==1,print1(n,", ")));
\\ Joerg Arndt, May 01 2013

from sympy import prime
from gmpy2 import powmod
A014127_list = [p for p in (prime(n) for n in range(1,10**7)) if powmod(3,p-1,p*p) == 1] # Chai Wah Wu, Dec 03 2014

Edited by Max Alekseyev, Oct 20 2010

Updated by Max Alekseyev, Jan 29 2012

A014545 Numbers k such that the k-th Euclid number A006862(k) = 1 + (Product of first k primes) is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865, 637491

Offset: 1

Eric W. Weisstein, Murray R. Bremner

			a(1) = 0 because the (empty) product of 0 primes is 1, plus 1 yields the prime 2.
prime(4413) = 42209 and Primorial(4413) + 1 = 42209# + 1 is a 18241-digit prime.
prime(13494) = 145823 and Primorial(13494) + 1 = 145823# + 1 is a 63142-digit prime.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 211, p. 61, Ellipses, Paris 2008.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 114.

C. K. Caldwell, Prime Pages: Database Search
C. K. Caldwell, Primorial Primes.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
Benny Lim, Prime Numbers Generated From Highly Composite Numbers, Parabola (2018) Vol. 54, Issue 3.
Eric Weisstein's World of Mathematics, Euclid Number
Eric Weisstein's World of Mathematics, Primorial Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A005234 (values of p such that 1 + product of primes <= p is prime).
Cf. A018239 (primorial plus 1 primes).
Cf. A002110, A006862, A057704.

P:= 1:
p:= 1:
count:= 0:
for n from 1 to 1000 do
  p:= nextprime(p);
  P:= P*p;
  if isprime(P+1) then
    count:= count+1;
    A[count]:= n;
  fi
od:
seq(A[i], i=1..count); # Robert Israel, Nov 04 2015

Flatten[Position[Rest[FoldList[Times,1,Prime[Range[180]]]]+1,?PrimeQ]] (* _Harvey P. Dale, May 04 2012 *) (* this program generates the first 9 positive terms of the sequence; changing the Range constant to 33237 will generate all 23 terms above, but it will take a long time to do so *)

is(n)=ispseudoprime(prod(i=1,n,prime(i))+1) \\ Charles R Greathouse IV, Mar 21 2013

P=1; n=0; forprime(p=1, 10^5, if(ispseudoprime(P+1), print1(n", ")); n=n+1; P*=p;) \\ Hans Loeblich, May 10 2019

a(n+1) = A000720(A005234(n)). - M. F. Hasler, May 31 2018

More terms from Labos Elemer
a(21) from Arlin Anderson (starship1(AT)gmail.com), Oct 20 2000
a(22)-a(23) from Eric W. Weisstein, Mar 13 2004 (based on information in A057704)
Offset and first term changed by Altug Alkan, Nov 27 2015
a(24) from Jeppe Stig Nielsen, Aug 08 2024
a(25) from Jeppe Stig Nielsen, Sep 01 2024
a(26) from Jeppe Stig Nielsen, Sep 24 2024
a(27) from Jeppe Stig Nielsen, Nov 10 2024
a(28) from Jeppe Stig Nielsen, Aug 21 2025

A250001 Number of arrangements of n circles in the affine plane.

Original entry on oeis.org

1, 1, 3, 14, 173, 16951

Offset: 0

Jon Wild, May 16 2014

Two circles are either disjoint or meet in two points. Tangential contacts are not allowed. A point belongs to exactly one or two circles. Three circles may not meet at a point. The circles may have different radii.
This is in the affine plane, rather than the projective plane.
Two arrangements are considered the same if one can be continuously changed to the other while keeping all circles circular (although the radii may be continuously changed), without changing the multiplicity of intersection points, and without a circle passing through an intersection point. Turning the whole configuration over is allowed.
Several variations are possible:
- straight lines instead of circles (see A241600).
- straight lines in general position (see A090338).
- curved lines in general position (see A090339).
- allow circles to meet tangentially but without multiple intersection points (begins 1, 5, ...); more terms are needed.
- again use circles, but allow multiple intersection points (also begins 1, 5, ...); more terms are needed.
- use ellipses rather than circles.
- a question from Walter D. Wallis: what if the circles must all have the same radius?
a(1)-a(5) computed by Jon Wild.
a(n) >= A000081(n+1) - Benoit Jubin, Dec 21 2014. More precisely, there are A000081(n+1) ways to arrange n circles if no two of them meet. - N. J. A. Sloane, May 16 2017
From Daniel Forgues, Aug 08-09 2015: (Start)
A representation for the diagrams in a250001.jpg (in the same order):
  a(1) =  1: {{2}};
  a(2) =  3: {{2, 3}, {2, 4}, {4, 6}};
  a(3) = 14: {{2, 4, 8}, {2, 3, 6}, {2, 3, 4}, {2, 3, 5}, {4, 6, 5},
    {4, 6, 15}, {2, 6, 9}, {4, 6, 12}, {2, 8, 12}, {30, 42, 70},
    {?, ?, ?}, {?, ?, ?}, {15, 21, 35}, {?, ?, ?}}.
In lexicographic order:
  a(3) = 14: {{2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 4, 8}, {2, 6, 9},
    {2, 8, 12}, {4, 6, 5}, {4, 6, 12}, {4, 6, 15}, {15, 21, 35},
    {30, 42, 70}, {?, ?, ?}, {?, ?, ?}, {?, ?, ?}}.
The smallest integers greater than 1 are used for the representation:
  (p_1)^(a_1)*...*(p_m)^(a_m), where
  0 <= a_i <= n, for 1 <= i <= m;
  (a_1)+...+(a_m) > 0.
Could the Venn diagram interpretation (of the k-wise, 1 <= k <= n, common divisors of k numbers from each subset) reveal a pattern?
Does this representation work for more complex diagrams? (End)
Once you get to n=5, geometric concerns mean that not all topologically-conceivable arrangements are actually circle-drawable.  My program enumerated 16977 conceivable arrangements of 5 pseudo-circles, and Christopher Jones and I together have figured out how to show that 26 of these arrangements are not actually circle-drawable. So it seems that a(5) = 16951. This entry will be updated soon, and there will be a new sequence for the number of topologically-conceivable arrangements. - Jon Wild, Aug 25 2016 [The counts in this comment were amended by Jon Wild on Aug 30 2016. I apologize for taking so long to make the corrections here. - N. J. A. Sloane, Jun 11 2017]
a(n) <= 7*13^(binomial(n,3) + binomial(n,2) + 3n - 1) is a (loose) upper bound, see Reddit link. I believe XkF21WNJ's reply shaves off a factor of 13^3 from this bound for all n > 1. - Linus Hamilton, Apr 14 2019
A good upper bound for a(6) is given in sequence A288559, which counts the arrangements of pseudo-circles, i.e. the topologically conceivable arrangements mentioned above, which are not all necessarily realizable with true circles. The number of arrangements of 6 pseudo-circles was found by Andrii Shportko and Jon Wild to be 17,552,169. - Jon Wild, Jun 03 2025
In A288559, a(5) included 26 non-circularizable pseudocircle arrangements, which generated in turn 132,546 6-pseudocircle descendants. These descendants must be excluded from A250001, which means that a tighter upper bound for A250001(6) is 17,419,623. - Andrii Shportko, Jun 06 2025

			a(2) = 3, because two circles can either be next to each other, overlap with two intersection points, or one may be located within the other (of larger radius). (As per the first comment, the limiting case where they touch in one point is [somewhat arbitrarily] excluded. This would add two more independent configurations, where one touched the other "from inside" or "from outside".) - _M. F. Hasler_, May 03 2025

Jon Wild, Posting to Sequence Fans Mailing List, May 15 2014.

Mohammad Arab, Creative proofs in combinations, arXiv:2112.08020 [math.CO], 2021-2022.
Andrew Cook and Luca Viganò, A Game Of Drones: Extending the Dolev-Yao Attacker Model With Movement, Proceedings of the 6th Workshop on Hot Issues in Security Principles and Trust (HotSpot 2020): Affiliated with Euro S&P 2020, IEEE Computer Science Press, Genova, Italy (2020).
Linus Hamilton, How many ways can circles overlap? - Numberphile, Reddit.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016. [Not directly related, but on a similar subject. - _N. J. A. Sloane_, Jan 20 2017]
N. J. A. Sloane, Illustration of a(2)=3 and a(3)=14
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, pp. 9, 21.
N. J. A. Sloane and Brady Haran, How many ways can circles overlap?, Numberphile video (2019)
Jon Wild, Illustrations of the 173 configurations of four circles
Jon Wild, Illustrations of the 112 connected configurations of four circles (Computer-generated svg file. To see it, save file, open it with a program - such as Chrome - that can handle svg files.)
Jon Wild, Figure showing relationship between A250001, A275923, A275924, and A288554 for n=3 circles
Jon Wild, Two inequivalent arrangements of 4 circles with same truth table of intersections.
Jon Wild, Email describing the arrangements of 4 circles with same truth table of intersections (see previous link)

Row sums of A261070.
Apart from first term, row sums of triangles A249752, A252158, A285996, A274776, A274777.
See A275923 and A275924 for the connected arrangements. See also A288554-A288568.
Cf. A241600, A090338, A090339, A048872, A048873, A003036, A132346.
Cf. also A000081, A274702, A274818, A274819, A285995, A287149.
Cf. A132101 (one-dimensional analog).

a(4) is 173, not 168. Corrected by Jon Wild, Aug 08 2015

A duplicate pair of configurations in an older file was spotted by Manfred Scheucher, Aug 13 2016. The pdf and svg files here are now correct.

A375161 Numbers k such that (23^k - 2^k)/21 is prime.

Original entry on oeis.org

5, 11, 197, 4159

Offset: 1

Robert Price, Aug 04 2024

The definition implies that k must be a prime.

a(5) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
P. Bourdelais, A Generalized Repunit Conjecture
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.

Select[Prime[Range[100]], PrimeQ[(23^# - 2^#)/21] &]

A007764 Number of nonintersecting (or self-avoiding) rook paths joining opposite corners of an n X n grid.

Original entry on oeis.org

1, 2, 12, 184, 8512, 1262816, 575780564, 789360053252, 3266598486981642, 41044208702632496804, 1568758030464750013214100, 182413291514248049241470885236, 64528039343270018963357185158482118, 69450664761521361664274701548907358996488

Offset: 1

David Radcliffe and Don Knuth

The length of the path varies.

			Suppose we start at (1,1) and end at (n,n). Let U, D, L, R denote steps that are up, down, left, right.
a(2) = 2: UR or RU.
a(3) = 12: UURR, UURDRU, UURDDRUU, URUR, URRU, URDRUU and their reflections in the x=y line.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, section 5.10, pp. 331-338.
Guttmann A J and Jensen I 2022 Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices Journal of Physics A: Mathematical and Theoretical 55 012345, (33pp) ; arXiv:2208.06744, Aug 2022.
D. E. Knuth, 'Things A Computer Scientist Rarely Talks About,' CSLI Publications, Stanford, CA, 2001, pages 27-28.
D. E. Knuth, The Art of Computer Programming, Section 7.1.4.
Shin-ichi Minato, The power of enumeration - BDD/ZDD-based algorithms for tackling combinatorial explosion, Chapter 3 of Applications of Zero-Suppressed Decision Diagrams, ed. T. Satsoa and J. T. Butler, Morgan & Claypool Publishers, 2014
Shin-ichi Minato, Counting by ZDD, Encyclopedia of Algorithms, 2014, pp. 1-6.
Netnews group rec.puzzles, Frequently Asked Questions (FAQ) file. (Science Section).

I. Jensen, H. Iwashita, and R. Spaans, Table of n, a(n) for n = 1..27 (I. Jensen computed terms 1 to 20, H. Iwashita computed 21 and 22, R. Spaans computed 23 to 25, and H. Iwashita computed 26 and 27)
M. Bousquet-Mélou, A. J. Guttmann and I. Jensen, Self-avoiding walks crossing a square, arXiv:cond-mat/0506341 [cond-mat.stat-mech], 2005.
Doi, Maeda, Nagatomo, Niiyama, Sanson, Suzuki, et al., Time with class! Let's count! [Youtube-animation demonstrating this sequence. In Japanese with English translation]
Steven R. Finch, Self-Avoiding-Walk Connective Constants
Anthony J. Guttmann and Iwan Jensen, The gerrymander sequence, or A348456, arXiv:2211.14482 [math.CO], 2022.
H. Iwashita, J. Kawahara, and S. Minato, ZDD-Based Computation of the Number of Paths in a Graph, TCS Technical Report, TCS-TR-A-12-60, Hokkaido University, September 18, 2012.
H. Iwashita, Y. Nakazawa, J. Kawahara, T. Uno, and S. Minato, Efficient Computation of the Number of Paths in a Grid Graph with Minimal Perfect Hash Functions, TCS Technical Report, TCS-TR-A-13-64, Hokkaido University, April 26, 2013.
I. Jensen, Series Expansions for Self-Avoiding Walks
Shin-ichi Minato, Power of Enumeration - Recent Topics on BDD/ZDD-Based Techniques for Discrete Structure Manipulation, IEICE Transactions on Information and Systems, Volume E100.D (2017), Issue 8, p. 1556-1562.
Shin-ichi Minato, Motivating Problems and Algorithmic Solutions, Algorithmic Foundations for Social Advancement, Springer, Singapore (2025), 17-29.
OEIS Wiki, Self-avoiding_walks
Kohei Shinohara, Atsuto Seko, Takashi Horiyama, Masakazu Ishihata, Junya Honda, and Isao Tanaka, Derivative structure enumeration using binary decision diagram, arXiv:2002.12603 [physics.comp-ph], 2020.
Ruben Grønning Spaans, C program
Jens Weise, Evolutionary Many-Objective Optimisation for Pathfinding Problems, Ph. D. Dissertation, Otto von Guercke Univ., Magdeburg (Germany, 2022), see p. 53.
Eric Weisstein's World of Mathematics, Self-Avoiding Walk

Main diagonal of A064298.

Cf. A064297, A271507, A001184, A000532.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A007764(n):
    if n == 1: return 1
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, n * n
    paths = GraphSet.paths(start, goal)
    return paths.len()
print([A007764(n) for n in range(1, 10)])  # Seiichi Manyama, Mar 21 2020

Computed to n=12 by John Van Rosendale in 1981
Extended to n=13 by Don Knuth, Dec 07 1995
Extended to n=20 by Mireille Bousquet-Mélou, A. J. Guttmann and I. Jensen
Extended to n=22 using ZDD technique based on Knuth's The Art of Computer Programming (exercise 225 in 7.1.4) by H. Iwashita, J. Kawahara, and S. Minato, Sep 18 2012
Extended to n=25 using state space compression (with rank/unrank) and dynamic programming (based in I. Jensen) by Ruben Grønning Spaans, Feb 22 2013
Extended to n=26 by Hiroaki Iwashita, Apr 11 2013
Extended to n=27 by Hiroaki Iwashita, Nov 18 2013

A375236 Numbers k such that (21^k - 2^k)/19 is prime.

Original entry on oeis.org

2, 3, 353, 751, 9587

Offset: 1

Robert Price, Aug 06 2024

The definition implies that k must be a prime.

a(6) > 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.

Select[Prime[Range[100]], PrimeQ[(21^# - 2^#)/19] &]

A005266 a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the largest prime factor of (b(n)-1).

Original entry on oeis.org

3, 2, 5, 29, 79, 68729, 3739, 6221191, 157170297801581, 70724343608203457341903, 46316297682014731387158877659877, 78592684042614093322289223662773, 181891012640244955605725966274974474087, 547275580337664165337990140111772164867508038795347198579326533639132704344301831464707648235639448747816483406685904347568344407941

Offset: 1

N. J. A. Sloane

Suggested by Euclid's proof that there are infinitely many primes.

a(15) requires completing the factorization: 13 * 67 * 14479 * 167197 * 924769 * 2688244927 * 888838110930755119 * 14372541055015356634061816579965403 * C211 where C211=6609133306626483634448666494646737799624640616060730302142187545405582531010390290502001156883917023202671554510633460047901459959959325342475132778791495112937562941066523907603281586796876335607258627832127303. - Sean A. Irvine, Nov 10 2009

R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.

R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.
S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)
Status of a(15) in factordb.com.

Cf. A000945, A000946, A005265.

Essentially the same as A084599.

with(numtheory):
a:= proc(n) option remember;
      `if`(n=1, 3, max(factorset(mul(a(i), i=1..n-1)-1)[]))
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Sep 26 2013

a[0] = 3; a[n_] := a[n] = Block[{p = Times @@ (a[#] & /@ Range[0, n - 1]) - 1}, FactorInteger[p][[-1, 1]]]; Array[a, 13] (* Robert G. Wilson v, Sep 26 2013 *)

a(14) from Joe K. Crump (joecr(AT)carolina.rr.com), Jul 26 2000

cp's OEIS Frontend

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