keyword:hard - cp's OEIS frontend

A092506 Prime numbers of the form 2^n + 1.

2, 3, 5, 17, 257, 65537

Offset: 1

Jorge Coveiro, Apr 05 2004

2 together with the Fermat primes A019434.
Obviously if 2^n + 1 is a prime then either n = 0 or n is a power of 2. - N. J. A. Sloane, Apr 07 2004
Numbers m > 1 such that 2^(m-2) divides (m-1)! and m divides (m-1)! + 1. - Thomas Ordowski, Nov 25 2014
From Jaroslav Krizek, Mar 06 2016: (Start)
Also primes p such that sigma(p-1) = 2p - 3.
Also primes of the form 2^n + 3*(-1)^n - 2 for n >= 0 because for odd n, 2^n - 5 is divisible by 3.
Also primes of the form 2^n + 6*(-1)^n - 5 for n >= 0 because for odd n, 2^n - 11 is divisible by 3.
Also primes of the form 2^n + 15*(-1)^n - 14 for n >= 0 because for odd n, 2^n - 29 is divisible by 3. (End)
Exactly the set of primes p such that any number congruent to a primitive root (mod p) must have at least one prime divisor that is also congruent to a primitive root (mod p). See the links for a proof. - Rafay A. Ashary, Oct 13 2016
Conjecture: these are the only solutions to the equation A000010(x)+A000010(x-1)=floor((3x-2)/2). - Benoit Cloitre, Mar 02 2018
For n > 1, if 2^n + 1 divides 3^(2^(n-1)) + 1, then 2^n + 1 is a prime. - Jinyuan Wang, Oct 13 2018
The prime numbers occurring in A003401. Also, the prime numbers dividing at least one term of A003401. - Jeppe Stig Nielsen, Jul 24 2019

Rafay A. Ashary, A Property of A092506
Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
Eric Weisstein's World of Mathematics, Fermat Prime
Eric Weisstein's World of Mathematics, Fermat Number

A019434 is the main entry for these numbers.

Cf. A000010, A000040, A000051, A000215, A003401.

Filtered(List([1..20],n->2^n+1),IsPrime); # Muniru A Asiru, Oct 25 2018

[2^n + 1 : n in [0..25] | IsPrime(2^n+1)]; // Vincenzo Librandi, Oct 14 2018

Select[2^Range[0,100]+1,PrimeQ] (* Harvey P. Dale, Aug 02 2015 *)

print1(2); for(n=0,9, if(ispseudoprime(t=2^2^n+1), print1(", "t))) \\ Charles R Greathouse IV, Aug 29 2016

A217320 Numbers n such that (13^n - 2^n)/11 is prime.

Original entry on oeis.org

3, 5, 31, 73, 89, 521, 3943, 8719

Offset: 1

Robert Price, Mar 18 2013

All terms are primes.

a(9) > 10^5.

Cf. A128028, A016054, A210506.

Select[ Prime[ Range[1, 100000] ], PrimeQ[ (13^# - 2^#)/11 ]& ]

is(n)=ispseudoprime((13^n-2^n)/11) \\ Charles R Greathouse IV, Jun 06 2017

A005265 a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the smallest prime factor of b(n)-1.

Original entry on oeis.org

3, 2, 5, 29, 11, 7, 13, 37, 32222189, 131, 136013303998782209, 31, 197, 19, 157, 17, 8609, 1831129, 35977, 508326079288931, 487, 10253, 1390043, 18122659735201507243, 25319167, 9512386441, 85577, 1031, 3650460767, 107, 41, 811, 15787, 89, 68168743, 4583, 239, 1283, 443, 902404933, 64775657, 2753, 23, 149287, 149749, 7895159, 79, 43, 1409, 184274081, 47, 569, 63843643

Offset: 1

N. J. A. Sloane

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

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.

Sean A. Irvine, Table of n, a(n) for n = 1..61
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.
OEIS wiki, OEIS sequences needing factors
S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)

Cf. A000945, A000946, A005266, A084599.

Essentially the same as A084598.

a :=n-> if n = 1 then 3 else numtheory:-divisors(mul(a(i),i = 1 .. n-1)-1)[2] fi:seq(a(n), n=1..15);
# Robert FERREOL, Sep 25 2019

lpf(n)=factor(n)[1,1] \\ better code exists, usually best to code in C and import
print1(A=3); for(n=2,99, a=lpf(A); print1(", "a); A*=a) \\ Charles R Greathouse IV, Apr 07 2020

A051783 Numbers k such that 3^k + 2 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 10, 14, 15, 24, 26, 36, 63, 98, 110, 123, 126, 139, 235, 243, 315, 363, 386, 391, 494, 1131, 1220, 1503, 1858, 4346, 6958, 7203, 10988, 22316, 33508, 43791, 45535, 61840, 95504, 101404, 106143, 107450, 136244, 178428, 361608, 504206, 1753088

Offset: 1

Jud McCranie, Dec 09 1999

From Farideh Firoozbakht and M. F. Hasler, Dec 06 2009: (Start)
If Q is a perfect number such that gcd(Q, 3(3^a(n) + 2)) = 1, then x = 3^(a(n) - 1)*(3^a(n) + 2)*Q is a solution of the equation sigma(x) = 3(x - Q). This is a result of the following theorem:
Theorem: If Q is a (q-1)-perfect number for some prime q, then for all integers t, the equation sigma(x) = q*x - (2t+1)*Q has the solution x = q^(k-1)*p*Q whenever k is a positive integer such that p = q^k + 2t is prime, gcd(q^(k-1), p) = 1 and gcd(q^(k-1)*p,Q) = 1.
Note that by taking t = -1/2(m*q+1), this theorem gives us some solutions of the equation sigma(x) = q *(x + m*Q). See comment lines of the sequence A058959. (End)
No further terms < 200000. - Ray Chandler, Jul 31 2011
A090649 implies that 361608 is a member of this sequence. - Robert Price, Aug 18 2014
No further terms < 320000. - Luke W. Richards, Mar 04 2018
a(45) and a(46) are probable primes because a primality certificate has not yet been found. They have been verified PRP with mprime. - Luke W. Richards, May 04 2018
No further terms < 1300000. - Luke W. Richards, May 17 2018
No further terms < 1400000. - Luke W. Richards, Jul 28 2020
Conjecture: The number n = 3^k + 2 is prime if and only if 2^((n-1)/2) == -1 (mod n). - Maheswara Rao Valluri, Jun 01 2020. [Note that this is an if and only if assertion, so it does not follow from Fermat's Little Theorem. - N. J. A. Sloane, Sep 07 2020]

			3^8 + 2 = 6563 is prime, so 8 is in the sequence.
3^26 + 2 = 2541865828331, a prime number, so 26 is in the sequence.

F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri and Renaud Lifchitz, PRP Records.

Cf. A057735, A087885, A014224, A058959.

A051783 = Select[Range[0, 20000], PrimeQ[3^# + 2] &]

is(n)=ispseudoprime(3^n+2) \\ Charles R Greathouse IV, Mar 21 2013

{4346, 6958, 7203} from David J. Rusin, Sep 29 2000
10988 from Ray Chandler, Nov 21 2004
{22316, 33508} found by Henri Lifchitz, Sep-Oct 2002
{43791, 45535, 61840} found by Henri Lifchitz, Oct-Nov 2004
95504 found by Wojciech Florek Dec 15 2005. - Alexander Adamchuk, Mar 02 2008
Edited by N. J. A. Sloane, Dec 19 2009
{101404, 106143, 107450, 136244} from Mike Oakes, Nov 2009
178428 from Ray Chandler, Jul 29 2011
a(45)-a(46) from Luke W. Richards, May 04 2018
a(47) from Paul Bourdelais, Mar 29 2022

A204940 Numbers n such that (23^n - 1)/22 is prime.

Original entry on oeis.org

5, 3181, 61441, 91943, 121949, 221411

Offset: 1

Robert Price, Jan 20 2012

No other terms < 100000.

H. Lifchitz, Mersenne and Fermat primes field

Cf. A028491, A004061, A004062, A004063, A004023, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A128000, A098438, A128002, A128003, A128004, A128005.

Select[Prime[Range[100]], PrimeQ[(23^#-1)/22]&]

is(n)=ispseudoprime((23^n-1)/22) \\ Charles R Greathouse IV, Jun 13 2017

a(5)=121949 corresponds to a probable prime discovered by Paul Bourdelais, Oct 19 2017

a(6)=221411 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020

A001606 Indices of prime Lucas numbers.

Original entry on oeis.org

0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849

Offset: 1

N. J. A. Sloane

Some of the larger entries may only correspond to probable primes.
Since (as noted under A000032) L(n) divides L(mn) whenever m is odd, L(n) cannot be prime unless n is itself prime, or else n contains no odd divisor, i.e., is a power of 2. Potential divisors of L(n) must satisfy certain linear forms dependent upon the parity of n, as shown in Vajda (1989), p. 82 (with a slight typographical error in the proof). - John Blythe Dobson, Oct 22 2007
Powers of 2 in this sequence are 2, 4, 8, 16; for 5 <= m <= 24, L(2^m) is composite; no factors of L(2^m) are known for m = 25, 26, 27, 29, 32, 33... (See Link section). - Serge Batalov, May 30 2017
2316773 is in the sequence, but its position is not yet defined. L(2316773) is a 484177-digit PRP. - Serge Batalov, Jun 11 2017

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 246.
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. Vajda, Fibonacci and Lucas numbers and the Golden Section: Theory and Applications. Chichester: Ellis Horwood Ltd., 1989.

J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
David Broadhurst, Lucas record follows Fibonacci, Yahoo! group "primenumbers", Apr 26 2001.
David Broadhurst, Lucas record follows Fibonacci [Cached copy].
C. K. Caldwell, The Prime Glossary, Lucas prime.
H. Dubner and W. Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417-427.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.
Blair Kelly, Factorizations of Lucas numbers
Alex Kontorovich and Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
H. Lifchitz and R. Lifchitz, PRP Top Records, L(n).
Mersenneforum, A collection of factors of L(2^m).
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
The Prime Database, V(81671), submitted Sep. 7, 2013, updated May 20, 2023.
Lawrence Somer and Michal Křížek, On Primes in Lucas Sequences, Fibonacci Quart. 53 (2015), no. 1, 2-23.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Cf. A000032, A000204, A001605, A005479.
Cf. A080327 (n for which Lucas(n) and Fibonacci(n) are both prime).
Subsequence of A076697 (indices for which gpf(A000032(n)) sets a new record).

Reap[For[k = 0, k < 20000, k++, If[PrimeQ[LucasL[k]], Print[k]; Sow[k]]] ][[2, 1]] (* Jean-François Alcover, Feb 27 2016 *)

is(n)=ispseudoprime(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Apr 24 2015

4 more terms from David Broadhurst, Jun 08 2001
More terms from T. D. Noe, Feb 15 2003 and Mar 04 2003; see link to The Prime Glossary.
387433, 443609, 532277 and 574219 found by Renaud Lifchitz, contributed by Eric W. Weisstein, Nov 29 2005
616787, 631181, 637751, 651821, 692147 found by Henri Lifchitz, circa Oct 01 2008, contributed by Alexander Adamchuk, Nov 28 2008
901657 and 1051849 found by Renaud Lifchitz, circa Nov 2008 and Mar 2009, contributed by Alexander Adamchuk, May 15 2010
1 more term from Serge Batalov, Jun 11 2017

A002827 Unitary perfect numbers: numbers k such that usigma(k) - k = k.

Original entry on oeis.org

6, 60, 90, 87360, 146361946186458562560000

Offset: 1

N. J. A. Sloane

d is a unitary divisor of k if gcd(d,k/d)=1; usigma(k) is their sum (A034448).
The prime factors of a unitary perfect number (A002827) are the Higgs primes (A057447). - Paul Muljadi, Oct 10 2005
It is not known if a(6) exists. - N. J. A. Sloane, Jul 27 2015
Frei proved that if there is a unitary perfect number that is not divisible by 3, then it is divisible by 2^m with m >= 144, it has at least 144 distinct odd prime factors, and it is larger than 10^440. - Amiram Eldar, Mar 05 2019
Conjecture: Subsequence of A083207 (Zumkeller numbers). Verified for all present terms. - Ivan N. Ianakiev, Jan 20 2020

			Unitary divisors of 60 are 1,4,3,5,12,20,15,60, with sum 120 = 2*60.
146361946186458562560000 = 2^18 * 3 * 5^4 * 7 * 11 * 13 * 19 * 37 * 79 * 109 * 157 * 313.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B3.
F. Le Lionnais, Les Nombres Remarquables. Paris: Hermann, p. 59, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section III.45.1.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 147-148.

H. A. M. Frei, Über unitar perfekte Zahlen, Elemente der Mathematik, Vol. 33, No. 4 (1978), pp. 95-96.
Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
M. V. Subbarao, Letter to N. J. A. Sloane, Feb 18 1974
M. V. Subbarao, T. J. Cook, R. S. Newberry and J. M. Weber, On unitary perfect numbers, Delta, 3 (No. 1, 1972), 22-26.
G. Villemin's Almanac of Numbers, Nombres Unitairement Parfaits
C. R. Wall, Letter to P. Hagis, Jr., Jan 13 1972
C. R. Wall, The fifth unitary perfect number, Canad. Math. Bull., 18 (1975), 115-122.
C. R. Wall, On the largest odd component of a unitary perfect number, Fib. Quart., 25 (1987), 312-316.
Eric Weisstein's World of Mathematics, Unitary Perfect Number.
Wikipedia, Unitary perfect number

Cf. A034460, A034448, A057447.
Subsequence of the following sequences: A003062, A290466 (seemingly), A293188, A327157, A327158.
Gives the positions of ones in A327159.

usnQ[n_]:=Total[Select[Divisors[n],GCD[#,n/#]==1&]]==2n; Select[Range[ 90000],usnQ] (* This will generate the first four terms of the sequence; it would take a very long time to attempt to generate the fifth term. *) (* Harvey P. Dale, Nov 14 2012 *)

is(n)=sumdivmult(n, d, if(gcd(d, n/d)==1, d))==2*n \\ Charles R Greathouse IV, Aug 01 2016

If m is a term and omega(m) = A001221(m) = k, then m < 2^(2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

A002863 Number of prime knots with n crossings.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, 46972, 253293, 1388705, 8053393, 48266466, 294130458

Offset: 1

N. J. A. Sloane

Prime knot: a nontrivial knot which cannot (as a composite knot can) be written as the knot sum of two nontrivial knots. - Jonathan Vos Post, Apr 30 2011

For convenience, many references and links related to the enumeration of knots are collected here, even if they do not explicitly refer to this sequence.
C. C. Adams, The Knot Book, Freeman, NY, 2001; see p. 33.
C. Cerf, Atlas of oriented knots and links, Topology Atlas 3 no. 2 (1998).
Peter R. Cromwell, Knots and Links, Cambridge University Press, 2004, pp. 209-211.
Martin Gardner, The Last Recreations, Copernicus, 1997, 67-84.
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).
P. G. Tait, Scientific Papers, Cambridge Univ. Press, Vol. 1, 1898, Vol. 2, 1900, see Vol. 1, p. 345.
M. B. Thistlethwaite, personal communication.

For convenience, many references and links related to the enumeration of knots are collected here, even if they do not explicitly refer to this sequence.
D. Bar-Natan, The Hoste-Thistlethwaite Table of 11 Crossing Knots
D. J. Broadhurst and D. Kreimer, Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops, arXiv:hep-th/9609128, 1996; Phys. Lett. B 393, No.3-4, 403-412 (1997).
B. Burton, The next 350 million knots, 36th International Symposium on Computational Geometry (SoCG 2020) (S. Cabello, D.Z. Chen, eds.), Leibniz Int. Proc. Inform., vol. 164, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2020, pp. 25:1-25:17.
Alain Caudron, Classification des noeuds et des enlacements (Thèse et additifs), Univ. Paris-Sud, 1989 [Scanned copy, included with permission. But also see the Perko links below.]
J. H. Conway, An enumeration of knots and links and some of their algebraic properties, 1970. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) pp. 329-358 Pergamon, Oxford.
S. R. Finch, Knots, links and tangles, Aug 08 2003. [Cached copy, with permission of the author]
Ortho Flint, Bruce Fontaine and Stuart Rankin, Enumerating the prime alternating links, preprint, 2007.
Ortho Flint and Stuart Rankin, Enumerating the prime alternating links, Journal of Knot theory and its Ramifications, 13 (2004), 151-173.
C. Giller, A family of links and the Conway calculus, Trans. American Math Soc., 270 (1982) 75-109.
Jeremy Green, A Table of Virtual Knots, 2004.
Hermann Gruber, Atlas of Rational Knots. [dead link]
J. Hoste, M. B. Thistlethwaite and J. Weeks, The First 1,701,936 Knots, Math. Intell., 20, 33-48, Fall 1998.
Jim Hoste, The Enumeration and Classification of Knots and Links, in Handbook of Knot Theory, William W. Menasco and Morwen B. Thistlethwaite, Editors, Elsevier, 2015.
S. Jablan, L. H. Kauffman, and P. Lopes, The delunification process and minimal diagrams, Topology Appl., 193 (2015), 270-289, #5531; see also, arXiv:1406.2378 [math.GT], 2014.
Knot Atlas, The Knot Atlas. Includes: The Rolfsen Table of knots with up to 10 crossings, The Hoste-Thistlethwaite Table of 11 Crossing Knots, The Thistlethwaite Link Table, The 36 Torus Knots with up to 36 Crossings, and The Mathematica Package KnotTheory.
Knotilus web site, Knotilus [dead link]
W. B. R. Lickorish and K. C. Millett, The new polynomial invariants of knots and links, Math. Mag. 61 (1988), no. 1, 3-23.
C. Livingston and A. H. Moore, KnotInfo: Table of Knot Invariants.
Andrei Malyutin, On the question of genericity of hyperbolic knots, arXiv preprint arXiv:1612.03368 [math.GT], 2016.
K. A. Perko, Jr., Abstract for Talk, 1973
K. A. Perko, Jr., On covering spaces of knots, Glasnik Mathematicki, Tom 9 (29) No. 1 (1974), 141-145. (Annotated scanned copy)
K. A. Perko, Jr., On the classification of knots, Proc. Amer. Math. Soc., 45 (1974), 262-266. (Annotated scanned copy)
K. A. Perko, Jr., Letters to N. J. A. Sloane 1974-1977
K. A. Perko, Jr., Primality of certain knots, In Topology Proceedings, vol. 7, no. 1, pp. 109-118. Auburn University Mathematics Department and the Institute for Medicine and Mathematics at Ohio University, 1982.
K. A. Perko, Jr., On ninth order knottiness, Preprint (N. D.)
K. A. Perko, Jr., Caudron's 1979 Knot Table, 2015 [Included with permission. See next link for list of errors.]
K. A. Perko, Jr., Errors in Alain Caudron's 1989 thesis
K. A. Perko, Jr., Review of Jablan-Kauffman-Lopes (2015)
Stuart Rankin, Knot Theory Preprints of Ortho Flint Smith and Stuart Rankin
S. Rankin and O. Flint Knot theory web page.
Stuart Rankin and Ortho Flint Smith, Enumerating the Prime Alternating Links, arXiv:math/0211451 [math.GT], 2002
Stuart Rankin, Ortho Flint Smith and John Schermann, Enumerating the Prime Alternating Knots, Part I, arXiv:math/0211346 [math.GT], 2002.
Stuart Rankin, Ortho Flint Smith and John Schermann, Enumerating the Prime Alternating Knots, Part II, arXiv:math/0211348 [math.GT], 2002.
Stuart Rankin, Ortho Flint Smith and John Schermann, Enumerating the Prime Alternating Knots, Part I, Journal of Knot Theory and its Ramifications, 13 (2004), 57-100.
Stuart Rankin, Ortho Flint Smith and John Schermann, Enumerating the Prime Alternating Knots, Part II, Journal of Knot Theory and its Ramifications, 13 (2004), 101-149.
R. G. Scharein, Number of Prime Links
Silvia Sconza and Arno Wildi, Knot-based Key Exchange protocol, Cryptology ePrint Archive (2024), Art. No. 2024/471. See Table 2, p. 15.
N. J. A. Sloane, Illustration of initial terms
P. G. Tait, The first seven orders of knottiness [Annotated scan of Plate VI]
M. B. Thistlethwaite, Home Page
M. B. Thistlethwaite, Numbers of alternating knots and links with up to 19 crossings
M. B. Thistlethwaite, Knot tabulations and related topics, Aspects of topology, 1-76, London Math. Soc. Lecture Note Ser., 93, Cambridge Univ. Press, Cambridge-New York, 1985.
Daniel Tubbenhauer and Victor Zhang, Big data comparison of quantum invariants, arXiv:2503.15810 [math.GT], 2025. See p. 4.
S. D. Tyurina, Diagram invariants of knots and the Kontsevich integral, J. Math. Sci. 134 (2) (2006) pp. 2017-2071.
University of Western Ontario Student Beowulf Initiative, Project: Prime Knots
Eric Weisstein's World of Mathematics, Knot.
Eric Weisstein's World of Mathematics, Prime Knot.
Eric Weisstein's World of Mathematics, Alternating Knot.
Eric Weisstein's World of Mathematics, Prime Link
R. G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993
Index entries for sequences related to knots

Cf. A002864, A086825.

Cf. A051766, A051767, A051768, A051769, A052400.

a(n) = A051766(n) + A051769(n) + A051767(n) + A051768(n) + A052400(n). - Andrew Howroyd, Oct 15 2020

This is stated incorrectly in CRC Standard Mathematical Tables and Formulae, 30th ed., first printing, 1996, p. 320.
Terms from Hoste et al. added by Eric W. Weisstein
Consolidated references and links on enumeration of knots into this entry, also created entry for knots in Index to OEIS. - N. J. A. Sloane, Aug 25 2015
a(17)-a(19) computed by Benjamin Burton, added by Alex Klotz, Jun 21 2021
a(17)-a(19) computed by Benjamin Burton corrected by Andrey Zabolotskiy, Nov 25 2021

A062587 Numbers k such that 21^k - 20^k is prime.

Original entry on oeis.org

2, 19, 41, 43, 337, 479, 9127, 37549, 44017, 59971, 128327, 176191, 193601

Offset: 1

Mike Oakes, May 18 2001, May 19 2001

Terms greater than 1000 may correspond to (unproven) strong pseudoprimes.

Top probable primes of the form 21^p-20^p

Cf. A000043, A057468, A059801, A059802, A062572-A062666.

is(n)=ispseudoprime(21^n-20^n) \\ Charles R Greathouse IV, Jun 13 2017

a(8) from Jean-Louis Charton, Feb 29 2012
a(9) and a(10) from Robert Price, May 30 2012
Edited by M. F. Hasler, Sep 16 2013
a(11) added by Jean-Louis Charton, Nov 24 2014
a(12) added by Jean-Louis Charton, Feb 05 2015
a(13) added by Jean-Louis Charton, Feb 18 2015

A119714 a(n) is the least k such that the remainder when 8^k is divided by k is n.

Original entry on oeis.org

7, 3, 5, 6, 39, 58, 7342733, 9, 36196439, 18, 501, 26, 13607, 249, 119, 20, 33, 25, 866401, 22, 533, 35, 185, 50, 196673, 27, 1843, 36, 69, 34, 551, 55, 3773365, 110, 159, 116, 355, 237, 8401, 52, 471, 81815, 85, 261, 11783479, 3258, 93, 92, 1885511821439

Offset: 1

Ryan Propper, Jun 12 2006

a(61) = 1802190094793 = 11 * 59 * 17839 * 155663. - Hagen von Eitzen, Jul 28 2009

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, A127816, A119715, A127817, A127818, A127819, A127820, A127821.

Do[k = 1; While[PowerMod[8, k, k] != n, k++ ]; Print[k], {n, 48}]
t = Table[0, {10000}]; k = 1; lst = {}; While[k < 4300000000, a = PowerMod[8, k, k]; If[ a<10001 && t[[a]]==0, t[[a]]=k; Print[{a,k}]]; k++ ]; t (* Mathematica coding extended to reflect the new search limits as posted in the a-file by Robert G. Wilson v, Jul 17 2009 *)

a(49) from Hagen von Eitzen, Jul 24 2009

cp's OEIS Frontend

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