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Integer Sequence Primes

Just as many interesting integer sequences can be defined and their properties studied, it is often of interest to additionally determine which of their elements are prime. The following table summarizes the indices of the largest known prime (or probable prime) members of a number of named sequences.

sequenceOEISn_(max)digitsdiscoverersearch limitcomments
alternating factorialA00127259961260448M. Rodenkirch (Sep. 18, 2017)100000 (M. Rodenkirch, Dec. 15, 2017)finite sequence; largest certified prime has index 661; the rest are probable primes
Apéry-constant primeA119334141141E. W. Weisstein (May 14, 2006)9089 (E. W. Weisstein, Mar. 22, 2008)status unknown
Apéry number A_nA092825662410136E. W. Weisstein (Mar. 2004)10^4 (E. W. Weisstein, Mar. 2004)probable prime
Apéry number B_n87E. W. Weisstein (Mar. 2004)1103800 (E. W. Weisstein, Mar. 16, 2004)proven prime (trivial)
Bell numberA0511302841653930447 (E. W. Weisstein, Apr. 23, 2006)proven prime by I. L. Canestro in 2004
Belphegor primeA232448181298362601S. Batalov (Nov. 15, 2014)?status unknown
Bernoulli number B_n numeratorA0921324222E. W. Weisstein (Mar. 2004)101114 (E. W. Weisstein, Mar. 21, 2009)proven prime
Catalan-constant primeA1183282547725477E. W. Weisstein (May. 5, 2006)166896 (E. W. Weisstein, Feb. 8, 2016)proven prime (trivial)
Catalan-Mersenne number c_n439Catalan (1876)c_5 has no prime factor less than 10^(51)Noll; private correspondence with C. K. Caldwell, Aug. 10, 2003; proven prime
central trinomial coefficient (n; 0)_242J. Vos Post (Feb. 2005)203661 (E. W. Weisstein, Oct. 30, 2015)proven prime (trivial)
central trinomial coefficient cousin i^nC_n^((-n))(-1/2i)A1128743187711139E. W. Weisstein (Feb. 25, 2005)10^5 (E. W. Weisstein, Mar. 7, 2005)status unknown
Champernowne-constant primeA0716203773537735E. W. Weisstein (Jul. 15, 2013)105867 (E. W. Weisstein, Oct. 9, 2015)status unknown
complementary Bell numberA1180187231266E. W. Weisstein (Apr. 10, 2006)32460 (E. W. Weisstein, Aug. 30, 2006)proven prime
Consecutive Number Sequences of cubic numbers--31152 (E. W. Weisstein, Oct. 9, 2015)none known
Consecutive Number Sequences of Fibonacci numbers441891 (E. Weisstein, Jul. 28, 2016)proven prime (trivial)
Consecutive Number Sequences of odd numbersA0460362570972537369 (E. W. Weisstein, Oct. 9, 2015)probable prime
Consecutive Number Sequences of square numbers3333432 (E. W. Weisstein, Oct. 9, 2015)none known
Consecutive Number Sequences of triangular numbers6935177 (E. W. Weisstein, Oct. 9, 2015)proven prime (trivial)
Consecutive Number Sequences of reverse integersA17602437765177719E. W. Weisstein (Apr. 6, 2010)46530 (E. W. Weisstein, Oct. 9, 2015)probable prime
Copeland-Erdős-constant primeA227530292447292447M. Rodenkirch (Dec. 11, 2015)500000 (M. Rodenkirch, Jun. 18, 2016)probable prime
Cullen number 2^nn+1A00584966798812010852PrimeGrid (Aug. 4, 2009)13705481 (PrimeGrid, Nov. 19, 2015)proven prime (http://primes.utm.edu/top20/page.php?id=6#records)
Delannoy number P_n(3)86E. W. Weisstein (Mar. 2004)7834059 (E. W. Weisstein, Feb. 28, 2006)proven prime (trivial)
double factorial prime n!!-1A007749145706344538S. Fukui (Apr. 21, 2015)??
double factorial prime n!!+1A080778123998288864S. Fukui (Jun. 5, 2015)??
double Mersenne number M_(M_n)73960 (G. Haworth 1983)proven prime; status of 61 is unknown, search being coordinated by T. Forbes
e-primeA064118155025155025E. W. Weisstein (Oct. 11, 2010)197760 (E. W. Weisstein, Feb. 8, 2016)proven prime
Euler-Mascheroni-constant primeA065815179849179849E. W. Weisstein (Jun. 1, 2011)206078 (E. W. Weisstein, Feb. 8, 2016)probable prime
Euler number prime |E_n|A1032345101062110000 (S. Plouffe, Apr. 13, 2024)proven prime (http://primes.utm.edu/primes/page.php?id=54561)
Euler primeA005846152...061398204D. Broadhurst (Feb. 11, 2014)?proven prime (http://primes.utm.edu/primes/page.php?id=111195)
factorial prime n!-1A0029822080031015843S. Fukui (Jul. 25, 2016)208003 (S. Fukui, Jul. 25, 2016)proven prime (http://primes.utm.edu/primes/page.php?id=121944)
factorial prime n!+1A002981150209712354PrimeGrid (Oct. 16, 2013)203550 (PrimeGrid, Jul. 25, 2016)prime proof underway (http://primes.utm.edu/primes/page.php?id=100445)
factorial sum sum_(k=1)^(n)k!^2A10028932841268128S. Balatov (Jul. 29, 2017)?probable prime; sequence must be finite
factorial sum sum_(k=1)^(n)k!^422E. W. Weisstein (Jul 16, 2017)completeproven prime
factorial sum sum_(k=1)^(n)k!^6A289946102972E. W. Weisstein (Jul 16, 2017)completeprobable prime
factorial sum sum_(k=1)^(n)k!^823E. W. Weisstein (Jul 16, 2017)completefinite sequence; all terms proven prime
factorial sum sum_(k=1)^(n)k!^(10)25252E. W. Weisstein (Jul 16, 2017)completefinite sequence; all terms proven prime
Fermat prime F_n4532finite sequence; all terms proven prime (trivial)
Fibonacci prime F_nA0016053244369678033H. Lifchitz (Nov. 2017)?probable prime; the largest proven prime has index 104911 (http://primes.utm.edu/top20/page.php?id=39#records)
fibonorial prime n!_F-1A0597091520R. G. Wilson (Feb. 7, 2001)800 (E. W. Weisstein, Dec. 8, 2009)proven prime (trivial)
fibonorial prime n!_F+1A05340828276R. G. Wilson (Jan. 8, 2000)800 (E. W. Weisstein, Dec. 8, 2009)proven prime (trivial)
Glaisher-constant primeA1184201969219692E. W. Weisstein (Apr. 29, 2006)129009 (E. W. Weisstein, Mar. 1, 2011)probable prime
Golomb-Dickman constant primeA17497425082508E. W. Weisstein (Apr. 3, 2010)15000 E. W. Weisstein, Jul. 25, 2013)probable prime
harmonic number H_n numeratorA0569038165835479E. W. Weisstein (May 12, 2009)81780 (E. W. Weisstein, May 13, 2009)status unknown
irregular primeA000928-numer(B_(22808)/22808)71290T. D. Noe (Sep. 28, 2005)probable prime; largest proven prime is 6B_(4306)/2153 (http://primes.utm.edu/primes/page.php?id=87451)
Khinchin-constant primeA1183272843328433E. W. Weisstein (Apr. 27, 2006)109843 (E. W. Weisstein, Mar. 21, 2009)probable prime
left factorial (L!n)/2A1006141222144640E. W. Weisstein (Oct. 19, 2006)12227 (E. W. Weisstein, Oct. 19, 2006)status unknown
Lucas prime L_nA0016061051849219824R. Lifchitz (Mar. 2009)probable prime; the largest proven prime has index 56003 (http://primes.utm.edu/primes/page.php?id=77992)
Mersenne prime M_nA00004313627984141024320L. Durant/GIMPS (Oct. 12, 2024)proven prime
Mills' prime b_nA05125413185052S. Batalov (Jul. 19, 2013)proven prime
Motzkin numberA0928313615E. W. Weisstein (Mar. 2004)303228 (E. W. Weisstein, Apr. 2, 2005)proven prime
natural logarithm of 2-constant primeA2282268997389973E. W. Weisstein (Oct. 27, 2015)90220 (E. W. Weisstein, Oct. 28, 2015)status unknown
natural logarithm of 10-constant primeA2282407506775067E. W. Weisstein (Oct. 10, 2015)77006 (E. W. Weisstein, Oct. 12, 2015)status unknown
near-square prime n^2-2A0288702^(376050)+1226405M. Rodenkirch(Feb. 2016)status unknown
near-square prime n^2+1A00557424518^(262144+1)1150678S. Scott (Mar. 12, 2008)generalized Fermat primeproven prime (http://primes.utm.edu/primes/page.php?id=84401)
near-square prime n^2+2A067201p_(near-square)110174D. Broadhurst (Apr. 8, 2006)proven prime (http://primes.utm.edu/primes/page.php?id=77539)
near-square prime (2^n-1)^2-2 ("Carol" prime)A091515653490393441M. Rodenkirch (Jun. 15, 2016)?status unknown
near-square prime (2^n+1)^2-2 ("Kynea" prime)A091513661478398250M. Rodenkirch (Jun. 19, 2016)?status unknown
NSW number S_nA113501139160106534R. Price (Dec. 7, 2018)subset of Pell-Lucas numbersprobable prime
Padovan sequence numberA11288272773488874E. W. Weisstein (Apr. 7, 2011)729586 (E. W. Weisstein, Apr. 10, 2011)status unknown
palindromic primeA002385p_(palindromic)474501S. Batalov (Nov. 2014)?proven prime (http://primes.utm.edu/top20/page.php?id=53#records)
partition function PA046063100000739635219E. W. Weisstein (Feb. 12, 2017)1000007862 (E. W. Weisstein, Feb. 12, 2017)status unknown
partition function QA03535981191852?M. Alekseyev (Jul. 10, 2015)10^8 (M. Alekseyev, Jul. 10, 2015)status unknown
Pell number P_nA0966509019734525T. D. Noe277096 (E. W. Weisstein, Oct. 9, 2015)probable prime; largest proven prime has index 13339 and 5106 digits (http://primes.utm.edu/primes/page.php?id=24572)
Pell-Lucas number Q_n/2A099088278321106535R. Price (Dec. 7, 2018)?probable prime; largest proven prime has index 9679 and 3705 digits (http://primes.utm.edu/primes/page.php?id=27783)
Perrin sequence numberA11288158113270970E. W. Weisstein (Feb. 25, 2011)601463 (E. W. Weisstein, Oct. 9, 2015)probable prime
phi-primeA0641199724197241E. W. Weisstein (Jun. 3, 2009)500000 (M. Rodenkirch, Jun. 20, 2017)status unknown
pi-primeA060421613373613373A. Bondrescu (May 29, 2016)?status unknown
Pierpont primeA0051093·2^(5082306)+11529928A. Brady (Apr. 4, 2009)?proven prime (http://primes.utm.edu/primes/page.php?id=87449)
power difference primeA072164791830870H. Lifchitz (2001)probable prime
primefree sequence inverted primesA10815618222038098E. W. Weisstein (Feb. 4, 2006)194202 (E. W. Weisstein, May 5, 2006)status unknown
primorial prime p_n#-1A0577042347251418398J. Winskill, PrimeGrid (Sep. 18, 2021)308253 (PrimeGrid, Aug. 10, 2024)proven prime (https://t5k.org/primes/page.php?id=132758)
primorial prime p_n#+1A0145453047231878843PrimeGrid (Jul. 27, 2024)310376 (PrimeGrid, Aug. 10, 2024)proven prime (http://primes.utm.edu/primes/page.php?id=138350)
Pythagoras-constant primeA1153771154011540E. W. Weisstein (Jan. 21, 2005)187336 (E. W. Weisstein, Feb. 8, 2016)proven prime
reflected pi-palindromic primeA11935120564111E. W. Weisstein (May 15, 2006)56755 (E. W. Weisstein, Mar. 21, 2009)status unknown
repunit prime (10^n-1)/9A00402381772078177207S. Batalov and R. Propper (May 8, 2021)10800000 (S. Batalov, Jul. 1, 2021)probable prime; largest proven prime has index 49081 (http://primes.utm.edu/top20/page.php?id=57)
Smarandache prime---10^6 (S. Batalov, Jun 15, 2018)none known
Smarandache-Wellin prime 2·3·5...p_nA04603514295719?1500000 (M. Rodenkirch, early 2016)probable prime
Soldner-constant primeA1224221947419474E. W. Weisstein (Sep. 6, 2006)21757 (E. W. Weisstein, Sep. 7, 2006)probable prime
Sophie Germain prime pA0053842618163402417·2^(1290000)-1388342PrimeGrid (Feb. 29, 2016)2559254540925 (PrimeGrid, Feb. 29, 2016)proven prime (http://primes.utm.edu/primes/page.php?id=121330)
super Catalan numberA092839216161E. W. Weisstein (Mar. 2004)584400 (E. W. Weisstein, Mar. 16, 2004)proven prime (trivial)
tau functionA135430773^(34960)555339N. Lygeros and O. Rozier (Sep. 2015)?probable prime
tetradic numberA068188p_(tetradic)180055D. Bedwell (Sep. 8, 2009)?proven prime (http://primes.utm.edu/top20/page.php?id=53#records)
tetranacci numberA10453415518244228E. W. Weisstein (Oct. 28, 2005)236965 (E. W. Weisstein, Mar. 29, 2009)probable prime
Thâbit ibn Kurrah prime 3·2^n-1A002235118957183580969PrimeGrid (Jun. 6, 2015)12078521 (PrimeGrid, Nov. 3, 2015)proven prime (http://primes.utm.edu/primes/page.php?id=120038)
Theodorus-constant primeA119344170657170657E. W. Weisstein (Aug. 18, 2013)193924 (E. W. Weisstein, Feb. 8, 2016)status unknown
tribonacci numberA0928359687825639K. Duszenko (Aug. 2003)291217 (E. W. Weisstein, Mar. 21, 2009)probable prime
twin primesA0013592996863034895·2^(1290000)+/-1388342PrimeGrid (Sep. 14, 2016)?proven prime (http://primes.utm.edu/top20/page.php?id=1#records)
Wagstaff primeA000978133725314025533R. Propper (Sep. 2013)?probable prime
Wall-Sun-Sun prime--1.45×10^(17) (PrimeGrid, Oct. 30, 2015)none known
Wieferich primeA001220351144.96×10^(17) (PrimeGrid, Oct. 30, 2015)proven prime (trivial)
Wilson primeA00754056332×10^(13) (Costa et al. 2012)proven prime (trivial)
Wolstenholme numberA1113544859042146T. D. Noe (Mar. 8, 2006)probable prime
Wolstenholme primeA0881642124679710^9 (McIntosh 2004)proven prime (trivial)
Woodall prime 2^nn-1A002234170166025122515D. Bertolotti (Mar. 22, 2018)16838832 (PrimeGrid, Mar. 22, 2018)proven prime (http://primes.utm.edu/primes/page.php?id=124539)

Here,

p_(near-square)=[30406^(12288)+889159(30406^(8192)+1)]^2+2
(1)
p_(palindromic)=10^(474500)+999·10^(237249)+1
(2)
p_(tetradic)=10^(180054)+8R_(58567)·10^(60744)+1.
(3)

See also

Consecutive Number Sequences, Integer Sequence, Prime Number, Probable Prime

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References

--. "The Great Smarandache PRPrime search." http://smarandache.ddns.net:1200/server_stats.html.Balatov, S. May 9, 2021. https://mersenneforum.org/showpost.php?p=578079&postcount=39.Balatov, S. May 9, 2021. https://mersenneforum.org/showthread.php?p=578120#post578120.Caldwell, C. and Gallot, Y. "On the Primality of n!+/-1 and 2×3×5×...×p+/-1." Math. Comput. 71, 441-448, 2002.Costa, E.; Gerbicz, R.; and Harvey, D. "A Search for Wilson Primes." 5 Dec 2012. http://arxiv.org/abs/1209.3436.Forbes, T. "MM61: A Search for a Factor of 2^(2^(61)-1)-1. Progress: 2 March 2004." http://www.ltkz.demon.co.uk/ar2/mm61prog.htm.Great Internet Prime Search: GIMPS. Finding World World Primes Since 1996. "List of Known Mersenne Prime Numbers." http://www.mersenne.org/primes/.Harvey, S. http://harvey563.tripod.com/Carol_Kynea.txt.Lifchitz, H. and Lifchitz, R. "PRP Records: Probable Primes Top 10000." http://www.primenumbers.net/prptop/prptop.php.McIntosh, R. email to Paul Zimmermann. 9 Mar 2004. http://www.loria.fr/~zimmerma/records/Wieferich.status.PrimeGrid. "BOINC Status: Subproject Status: LLR." http://www.primegrid.com/server_status_subprojects.php.PrimeGrid. http://www.primegrid.com/stats_321_llr.php.PrimeGrid. "Primorial Prime Search Project: Range Statistics." http://www.primegrid.com/stats_prs.php.Rodenkirch, M. "Alternating Factorials." Dec. 15, 2017. http://www.mersenneforum.org/showthread.php?p=474083#post474083.Rodenkirch, M. "Copeland-Erdos Constant Primes." May. 5, 2016. http://www.mersenneforum.org/showthread.php?p=433145#post433145.Rodenkirch, M. "Carol and Kynea Prime Search." http://www.mersenneforum.org/rogue/ckps.html.Rodenkirch, M. "Primes in phi (phi)." Apr. 24, 2017. http://www.mersenneforum.org/showthread.php?p=457389#post457389.Sloane, N. J. A. Sequences A000043, A000928, A001220, A001605, A001606, A002234, A002235, A002981/M0908, A002982/M2321, A004023, A005849/M5401, A007749, A014545, A014547, A035359, A046036, A046063, A046284, A051130, A057704, A071620, A080778, A088164, A091513, A091515, A092132, A092825, A092831, A092835, A092839, A104534, A108156, A112881, A112882, A174974, A176024, A227530, and A232448 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Integer Sequence Primes

Cite this as:

Weisstein, Eric W. "Integer Sequence Primes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IntegerSequencePrimes.html

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