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Integer Sequence Primes

Integer sequence primes are prime elements of integer sequences. The following table summarizes reported prime and probable prime records in named sequences. For indexed sequences, the index is given; for unindexed classes, the prime value is given. The number of decimal digits refers to the prime or probable prime itself. Search limits and dates refer to the individual reports listed.

The notation used in the table is

n_(near-square)=30406^(12288)+889159(30406^(8192)+1)
(1)
p_(palindromic)=10^(2718281)-5·10^(1631138)-5·10^(1087142)-1
(2)
p_(tetradic)=10^(180054)+8R_(58567)·10^(60744)+1,
(3)

where R_n=(10^n-1)/9 is a repunit.

sequenceOEISindex or valuedigitsdiscoverersearch limitcomments
alternating factorial primeA00127259961260448M. Rodenkirch (Sep. 18, 2017)100000 (M. Rodenkirch, Dec. 15, 2017)finite sequence; largest certified prime has index 661; larger reported terms are probable primes
Apéry-constant primeA119334141141E. W. Weisstein (May 14, 2006)10000 (H. P. Dale, Aug. 28, 2019)status unknown
Apéry number prime A_nA092825662410136E. W. Weisstein (Mar. 2004)10^4 (E. W. Weisstein, Mar. 2004)probable prime
Apéry number prime B_nA09282887E. W. Weisstein (Mar. 2004)1103800 (E. W. Weisstein, Mar. 16, 2004)proved prime; OEIS lists values, not indices
Bell numberA05113028416539n<100000 (M. Gouttenoire, Oct. 31, 2021)proven prime by I. L. Canestro in 2004
Belphegor primeA232448181298362601S. Batalov (Nov. 15, 2014)?proven prime (https://t5k.org/primes/page.php?id=118770)
Bernoulli number B_n numeratorA0921324222E. W. Weisstein (Mar. 2004)300000 (S. Batalov, Jun. 27, 2025)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)probable prime
Champernowne-constant primeA0716203773537735E. W. Weisstein (Jul. 15, 2013)105867 (E. W. Weisstein, Oct. 9, 2015)status unknown
Colbert number10223×2^(31172165)+19383761S. Peter/PrimeGrid (Oct. 31, 2016)proven prime (https://t5k.org/primes/page.php?id=122473)
complementary Bell numberA1180187231266E. W. Weisstein (Apr. 10, 2006)40968 (E. W. Weisstein, Mar. 21, 2009)probable prime
consecutive number sequences of cubic numbers--31152 (E. W. Weisstein, Oct. 9, 2009)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)proven prime (trivial)
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)84300 (T. Busby, Feb. 21, 2023)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)n<29274954 (PrimeGrid, Oct. 29, 2026)proven prime (https://t5k.org/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, 2016)?proven prime
double factorial prime n!!+1A080778220502541239S. Fukui (Jan. 17, 2017)?proven prime (https://t5k.org/primes/page.php?id=122753)
double Mersenne number M_(M_n)73960 (G. Haworth 1983); M_(M_(61)) checked to k=6·10^(17) (DoubleMersennes.org)proven prime; status of 61 is unknown
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|A1032345101062112130 (S. Plouffe, Apr. 15, 2024)proven prime (https://t5k.org/primes/page.php?id=54561)
Euler primeA005846152...061398204D. Broadhurst (submitted Feb. 12, 2013)?proven prime (https://t5k.org/primes/page.php?id=111195)
factorial prime n!-1A0029826327603395992R. Propper (Oct. 20, 2024)617909 (PrimeGrid, Jun. 5, 2026)proven prime (https://t5k.org/primes/page.php?id=138667)
factorial prime n!+1A0029814224292193027R. Propper (Feb. 21, 2022)603136 (PrimeGrid, Jun. 5, 2026)proven prime (https://t5k.org/primes/page.php?id=133604)
factorial sum sum_(k=1)^(n)k!^2A10028932841268128S. Balatov (Jul. 29, 2017)?finite sequence; probable prime
factorial sum sum_(k=1)^(n)k!^422E. W. Weisstein (Jul. 16, 2017)completefinite sequence; all terms proven prime
factorial sum sum_(k=1)^(n)k!^6A289947102972E. W. Weisstein (Jul. 16, 2017)completefinite sequence; probable 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_n4532all known terms proven prime; finiteness is unproved
Fibonacci prime F_nA001605119642992500391R. Propper and S. Batalov (2025)?probable prime; largest proven prime has index 201107 (https://t5k.org/top20/page.php?id=39#records)
fibonorial prime n!_F-1A0597091520R. G. Wilson (Feb. 7, 2001)1300 (M. S. Branicky, Aug. 17, 2024)proven prime (trivial)
fibonorial prime n!_F+1A0534082876R. G. Wilson (Jan. 8, 2000)1100 (R. Price, May 27, 2019)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 PrimePages record has 35160 digits (https://t5k.org/top20/page.php?id=26#records)
Khinchin-constant primeA1183272843328433E. W. Weisstein (Apr. 27, 2006)250000 (E. W. Weisstein, Oct. 17, 2026)probable prime
left factorial (L!n)/2A10061491743415436Ž. Mijajlović (Nov. 2025)197651 (Ž. Mijajlović, Nov. 26, 2025)reported probable prime (Mijajlović)
Lucas prime L_nA0016062316773484177S. Batalov (reported 2017)probable prime; largest proven prime has index 202667 (https://t5k.org/top20/page.php?id=48#records)
Mersenne prime M_nA00004313627984141024320L. Durant/GIMPS (Oct. 12, 2024)proven prime
Mills' prime f_nA051254151665461R. Propper and S. Batalov (Apr. 2024)probable prime; sequence membership conditional on earlier terms being prime (OEIS A108739)
Motzkin numberA0928313615E. W. Weisstein (Mar. 2004)20000000 (S. Batalov, Feb. 2, 2022)proven prime
natural logarithm of 2-constant primeA2282268997389973E. W. Weisstein (reported Oct. 28, 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^(695631)-1418812M. Rodenkirch (Jul. 16, 2016)proven prime (https://t5k.org/primes/page.php?id=121905)
near-square prime n^2+1A0055742524190^(1048576)13426224T. Greer/PrimeGrid (Oct. 2025)generalized Fermat numberproven prime (https://t5k.org/primes/page.php?id=141162)
near-square prime n^2+2A067201n_(near-square)110174D. Broadhurst (Apr. 8, 2006)proven prime (https://t5k.org/primes/page.php?id=77539)
near-square prime (2^n-1)^2-2 ("Carol" prime)A091515695631418812M. Rodenkirch (Jul. 16, 2016)?proven prime (https://t5k.org/primes/page.php?id=121905)
near-square prime (2^n+1)^2-2 ("Kynea" prime)A091513852770513419R. Propper (reported Jul. 2019)?proven prime (https://t5k.org/primes/page.php?id=126646)
NSW number S_nA113501276141211401R. Price (reported Dec. 2023)subset of Pell-Lucas numbersprobable prime; corresponds to Pell-Lucas number index 552283 (OEIS A099088)
Padovan primeA11288272773488874E. W. Weisstein (Apr. 7, 2011)729586 (E. W. Weisstein, Apr. 10, 2011)probable prime
palindromic primeA002385p_(palindromic)2718281R. Propper/S. Batalov (Aug. 6, 2024)?proven prime (https://t5k.org/primes/page.php?id=138383)
partition function PA046063300048483361012G. Childers (submitted Sep. 17, 2026)1000007862 (E. W. Weisstein, Feb. 12, 2017)proven prime (https://t5k.org/primes/page.php?id=142460)
partition function QA03535981191852?M. Alekseyev (Jul. 10, 2015)10^8 (M. Alekseyev, Jul. 10, 2015)status unknown
Pell prime P_nA0966509019734525T. D. Noe277096 (E. W. Weisstein, Oct. 9, 2015)probable prime; PrimePages lists a smaller PRP with index 13339 (https://t5k.org/primes/page.php?id=24572)
Pell-Lucas number Q_n/2A099088552283211401R. Price (reported Dec. 2023)?probable prime; PrimePages lists a smaller PRP with index 9679 (https://t5k.org/primes/page.php?id=27783)
Perrin prime P_nA1128811430138174654R. Propper (reported 2022)?probable prime
phi-primeA0641199724197241E. W. Weisstein (reported Jun. 4, 2009)500000 (M. Rodenkirch, Jun. 20, 2017)status unknown
pi-primeA060421613373613373A. Bondrescu (May 29, 2016)?probable prime; largest proven prime has index 16208
Pierpont primeA00510981·2^(20498148)+16170560R. Propper (Jun. 2023)?proven prime (https://t5k.org/primes/page.php?id=136165)
power difference primeA072164791830870H. Lifchitz (2001)n<25000 (T. D. Noe, Jun. 12, 2008)probable prime
primefree sequence inverted primesA10815618222038098E. W. Weisstein (Feb. 4, 2006)194202 (E. W. Weisstein, May 5, 2006)status unknown
primorial prime p_n#-1A0577044468952835864PrimeGrid (Aug. 18, 2024)?proven prime (https://t5k.org/primes/page.php?id=138455)
primorial prime p_n#+1A0145456374914151498PrimeGrid (Jun. 25, 2025)?proven prime (https://t5k.org/primes/page.php?id=140909)
Pythagoras-constant primeA1153771154011540E. W. Weisstein (reported Jan. 21, 2006)800000 (A. Bondrescu, Mar. 17, 2026)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 109297 (https://t5k.org/top20/page.php?id=57#records)
Smarandache prime---1200000 (WraithX, Mar. 23, 2025)none known
Smarandache-Wellin prime concatenation of the first n primesA04603514295719?4000000 (M. Rodenkirch, Nov. 29, 2025)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 (https://t5k.org/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 (https://t5k.org/top20/page.php?id=53#records)
tetranacci numberA10453415518244228E. W. Weisstein (Oct. 28, 2005)236965 (E. W. Weisstein, Mar. 21, 2009)probable prime
Thābit ibn Qurra prime 3·2^n-1A002235231578756971216S. Brown, PrimeGrid (Jun. 25, 2025)n<25952926 (PrimeGrid, Oct. 29, 2026)proven prime (https://t5k.org/primes/page.php?id=140896)
Theodorus-constant primeA119344170657170657E. W. Weisstein (Aug. 18, 2013)193924 (E. W. Weisstein, Feb. 8, 2016)status unknown
tribonacci numberA0928359687825639K. Duszenko (Aug. 2003)600000 (R. Price, Nov. 21, 2013)probable prime
twin primesA0013592996863034895·2^(1290000)+/-1388342PrimeGrid (Sep. 14, 2016)?proven prime (https://t5k.org/top20/page.php?id=1#records)
Wagstaff primeA000978151353974556209R. Propper (reported 2021)?probable prime
Wall-Sun-Sun prime--2^(64) (PrimeGrid, Dec. 2022)none known
Wieferich primeA001220351142^(64) (PrimeGrid, Dec. 2022)proven prime (trivial)
Wilson primeA00754056332×10^(13) (Costa et al. 2014)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)n<26877541 (PrimeGrid, Oct. 29, 2026)proven prime (https://t5k.org/primes/page.php?id=124539)

See also

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

Explore with Wolfram|Alpha

References

--. "The Great Smarandache PRPrime Search." http://smarandache.ddns.net:1200/server_stats.html.Balatov, S. May 9, 2021. https://www.mersenneforum.org/node/20483/page2#post782828.Balatov, S. May 9, 2021. https://www.mersenneforum.org/node/20483/page2#post782867.Barnes, G. "Kynea Primes for Bases <=1000." No Prime Left Behind. Updated Oct. 23, 2025. http://www.noprimeleftbehind.net/Kynea-primes-1000.htm.Batalov, S. "Mills Primes." GitHub. https://github.com/s-batalov/Mills_Primes.Batalov, S. and Propper, R. "Mills' Primes (PRPs) Series Extensions." 2024. https://github.com/s-batalov/Mills_Primes.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." Math. Comput. 83, 3071-3091, 2014. https://doi.org/10.1090/S0025-5718-2014-02800-7.DoubleMersennes.org. "Status of MM_p Where M_p Is a Mersenne Prime." https://www.doublemersennes.org/history.php.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 Record Primes Since 1996. "List of Known Mersenne Prime Numbers." https://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. https://members.loria.fr/PZimmermann/records/Wieferich.status.Mijajlović, Ž. "Kurepa Prime Numbers." Mathematical Institute of the Serbian Academy of Sciences and Arts. https://www.mi.sanu.ac.rs/novi_sajt/news/news_pages/Kurepa_prime_numbers.pdf.Numberphile. "The Most Wanted Prime Number." Featuring N. J. A. Sloane. Dec. 15, 2021. https://www.youtube.com/watch?v=vKlVNFOHJ9I.OEIS Foundation. Sequences related to Most Wanted Primes video. https://oeis.org/wiki/Index_to_OEIS:_Section_Mo#MWP.Prime k-Tuplets. "Exhaustive Search Table for Generalized (Carol-, Kynea-, Repunit-, Wagstaff-, Mersenne) Primes." Updated Oct. 15, 2026. https://www.pzktupel.de/Primetables/Table_EST_Gen_Primes.php.PrimeGrid. "BOINC Status: Subproject Status: LLR." https://www.primegrid.com/server_status_subprojects.php.PrimeGrid. https://www.primegrid.com/stats_321_llr.php.PrimeGrid. "Primorial Prime Search Project: Range Statistics." https://www.primegrid.com/stats_prs.php.Rodenkirch, M. "Copeland-Erdos Constant Primes." May 5, 2016. https://www.mersenneforum.org/node/15239/page2#post585416.Rodenkirch, M. "Alternating Factorials." Dec. 15, 2017. https://www.mersenneforum.org/node/16922#post641755.Rodenkirch, M. "Primes in phi (phi)." Apr. 24, 2017. https://www.mersenneforum.org/node/16417#post618391.Rodenkirch, M. "Smarandache-Wellin Primes." Nov. 11, 2025. https://www.mersenneforum.org/node/15124/page7#post1097782.Rodenkirch, M. "Carol and Kynea Prime Search." https://www.mersenneforum.org/rogue/ckps.html.Sloane, N. J. A. Sequences A000043, A000928, A000978, A001220, A001272, A001359, A001605, A001606, A002234, A002235, A002385, A002981/M0908, A002982/M2321, A004023, A005109, A005384, A005574, A005846, A005849/M5401, A007540, A007749, A014545, A014547, A028870, A035359, A046035, A046036, A046063, A046284, A051130, A051254, A053408, A056903, A057704, A059709, A060421, A064118, A064119, A065815, A067201, A068188, A071620, A072164, A080778, A088164, A091513, A091515, A092132, A092825, A092828, A092831, A092835, A092839, A096650, A099088, A100289, A100614, A103234, A104534, A108156, A108739, A111354, A112874, A112881, A112882, A113501, A115377, A118018, A118327, A118328, A118420, A119334, A119344, A119351, A122422, A135430, A174974, A176024, A227530, A228226, A228240, A232448, and A289947 in "The On-Line Encyclopedia of Integer Sequences."WraithX. "Smarandache Prime(s?)." Mersenne Forum, Mar. 23, 2025. https://www.mersenneforum.org/node/15124?p=1072727#post1072727.

Referenced on Wolfram|Alpha

Integer Sequence Primes

Cite this as:

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

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