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Date: 22-11-2019
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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.
sequence | OEIS | digits | discoverer | search limit | comments | |
alternating factorial | A001272 | 59961 | 260448 | M. 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 prime | A119334 | 141 | 141 | E. W. Weisstein (May 14, 2006) | 9089 (E. W. Weisstein, Mar. 22, 2008) | status unknown |
Apéry number | A092825 | 6624 | 10136 | E. W. Weisstein (Mar. 2004) | (E. W. Weisstein, Mar. 2004) | probable prime |
Apéry number | 8 | 7 | E. W. Weisstein (Mar. 2004) | 1103800 (E. W. Weisstein, Mar. 16, 2004) | proven prime (trivial) | |
Bell number | A051130 | 2841 | 6539 | 30447 (E. W. Weisstein, Apr. 23, 2006) | proven prime by I. L. Canestro in 2004 | |
Belphegor prime | A232448 | 181298 | 362601 | S. Batalov (Nov. 15, 2014) | ? | status unknown |
Bernoulli number numerator | A092132 | 42 | 22 | E. W. Weisstein (Mar. 2004) | 101114 (E. W. Weisstein, Mar. 21, 2009) | proven prime |
Catalan-constant prime | A118328 | 25477 | 25477 | E. W. Weisstein (May. 5, 2006) | 166896 (E. W. Weisstein, Feb. 8, 2016) | proven prime (trivial) |
Catalan-Mersenne number | 4 | 39 | Catalan (1876) | has no prime factor less than | Noll; private correspondence with C. K. Caldwell, Aug. 10, 2003; proven prime | |
central trinomial coefficient | 4 | 2 | J. Vos Post (Feb. 2005) | 203661 (E. W. Weisstein, Oct. 30, 2015) | proven prime (trivial) | |
central trinomial coefficient cousin | A112874 | 31877 | 11139 | E. W. Weisstein (Feb. 25, 2005) | (E. W. Weisstein, Mar. 7, 2005) | status unknown |
Champernowne-constant prime | A071620 | 37735 | 37735 | E. W. Weisstein (Jul. 15, 2013) | 105867 (E. W. Weisstein, Oct. 9, 2015) | status unknown |
complementary Bell number | A118018 | 723 | 1266 | E. 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 numbers | 4 | 4 | 1891 (E. Weisstein, Jul. 28, 2016) | proven prime (trivial) | ||
consecutive number sequences of odd numbers | A046036 | 2570 | 9725 | 37369 (E. W. Weisstein, Oct. 9, 2015) | probable prime | |
consecutive number sequences of square numbers | 3 | 3 | 33432 (E. W. Weisstein, Oct. 9, 2015) | none known | ||
consecutive number sequences of triangular numbers | 6 | 9 | 35177 (E. W. Weisstein, Oct. 9, 2015) | proven prime (trivial) | ||
consecutive number sequences of reverse integers | A176024 | 37765 | 177719 | E. W. Weisstein (Apr. 6, 2010) | 46530 (E. W. Weisstein, Oct. 9, 2015) | probable prime |
Copeland-Erdős-constant prime | A227530 | 292447 | 292447 | M. Rodenkirch (Dec. 11, 2015) | 500000 (M. Rodenkirch, Jun. 18, 2016) | probable prime |
Cullen number | A005849 | 6679881 | 2010852 | PrimeGrid (Aug. 4, 2009) | 13705481 (PrimeGrid, Nov. 19, 2015) | proven prime (http://primes.utm.edu/top20/page.php?id=6#records) |
Delannoy number | 8 | 6 | E. W. Weisstein (Mar. 2004) | 7834059 (E. W. Weisstein, Feb. 28, 2006) | proven prime (trivial) | |
double factorial prime | A007749 | 145706 | 344538 | S. Fukui (Apr. 21, 2015) | ? | ? |
double factorial prime | A080778 | 123998 | 288864 | S. Fukui (Jun. 5, 2015) | ? | ? |
double Mersenne number | 7 | 39 | 60 (G. Haworth 1983) | proven prime; status of 61 is unknown, search being coordinated by T. Forbes | ||
e-prime | A064118 | 155025 | 155025 | E. W. Weisstein (Oct. 11, 2010) | 197760 (E. W. Weisstein, Feb. 8, 2016) | proven prime |
Euler-Mascheroni-constant prime | A065815 | 179849 | 179849 | E. W. Weisstein (Jun. 1, 2011) | 206078 (E. W. Weisstein, Feb. 8, 2016) | probable prime |
Euler number prime | A103234 | 510 | 1062 | 69574 (S. Plouffe, Aug. 21, 2020) | proven prime (http://primes.utm.edu/primes/page.php?id=54561) | |
Euler prime | A005846 | 398204 | D. Broadhurst (Feb. 11, 2014) | ? | proven prime (http://primes.utm.edu/primes/page.php?id=111195) | |
factorial prime | A002982 | 208003 | 1015843 | S. Fukui (Jul. 25, 2016) | 208003 (S. Fukui, Jul. 25, 2016) | proven prime (http://primes.utm.edu/primes/page.php?id=121944) |
factorial prime | A002981 | 150209 | 712354 | PrimeGrid (Oct. 16, 2013) | 203550 (PrimeGrid, Jul. 25, 2016) | prime proof underway (http://primes.utm.edu/primes/page.php?id=100445) |
factorial sum | A100289 | 32841 | 268128 | S. Balatov (Jul. 29, 2017) | ? | probable prime; sequence must be finite |
factorial sum | 2 | 2 | E. W. Weisstein (Jul 16, 2017) | complete | proven prime | |
factorial sum | A289946 | 102 | 972 | E. W. Weisstein (Jul 16, 2017) | complete | probable prime |
factorial sum | 2 | 3 | E. W. Weisstein (Jul 16, 2017) | complete | finite sequence; all terms proven prime | |
factorial sum | 25 | 252 | E. W. Weisstein (Jul 16, 2017) | complete | finite sequence; all terms proven prime | |
Fermat prime | 4 | 5 | 32 | finite sequence; all terms proven prime (trivial) | ||
Fibonacci prime | A001605 | 3244369 | 678033 | H. Lifchitz (Nov. 2017) | ? | probable prime; the largest proven prime has index 104911 (http://primes.utm.edu/top20/page.php?id=39#records) |
fibonorial prime | A059709 | 15 | 20 | R. G. Wilson (Feb. 7, 2001) | 800 (E. W. Weisstein, Dec. 8, 2009) | proven prime (trivial) |
fibonorial prime | A053408 | 28 | 276 | R. G. Wilson (Jan. 8, 2000) | 800 (E. W. Weisstein, Dec. 8, 2009) | proven prime (trivial) |
Glaisher-constant prime | A118420 | 19692 | 19692 | E. W. Weisstein (Apr. 29, 2006) | 129009 (E. W. Weisstein, Mar. 1, 2011) | probable prime |
Golomb-Dickman constant prime | A174974 | 2508 | 2508 | E. W. Weisstein (Apr. 3, 2010) | 15000 E. W. Weisstein, Jul. 25, 2013) | probable prime |
harmonic number numerator | A056903 | 81658 | 35479 | E. W. Weisstein (May 12, 2009) | 81780 (E. W. Weisstein, May 13, 2009) | status unknown |
irregular prime | A000928 | 71290 | T. D. Noe (Sep. 28, 2005) | probable prime; largest proven prime is (http://primes.utm.edu/primes/page.php?id=87451) | ||
Khinchin-constant prime | A118327 | 28433 | 28433 | E. W. Weisstein (Apr. 27, 2006) | 109843 (E. W. Weisstein, Mar. 21, 2009) | probable prime |
left factorial | A100614 | 12221 | 44640 | E. W. Weisstein (Oct. 19, 2006) | 12227 (E. W. Weisstein, Oct. 19, 2006) | status unknown |
Lucas prime | A001606 | 1051849 | 219824 | R. Lifchitz (Mar. 2009) | probable prime; the largest proven prime has index 56003 (http://primes.utm.edu/primes/page.php?id=77992) | |
Mersenne prime | A000043 | 82589933 | 24862048 | GIMPS/P. Laroche/GIMPS (Dec. 7, 2018) | proven prime | |
Mills' prime | A051254 | 13 | 185052 | S. Batalov (Jul. 19, 2013) | proven prime | |
Motzkin number | A092831 | 36 | 15 | E. W. Weisstein (Mar. 2004) | 303228 (E. W. Weisstein, Apr. 2, 2005) | proven prime |
natural logarithm of 2-constant prime | A228226 | 89973 | 89973 | E. W. Weisstein (Oct. 27, 2015) | 90220 (E. W. Weisstein, Oct. 28, 2015) | status unknown |
natural logarithm of 10-constant prime | A228240 | 75067 | 75067 | E. W. Weisstein (Oct. 10, 2015) | 77006 (E. W. Weisstein, Oct. 12, 2015) | status unknown |
near-square prime | A028870 | 226405 | M. Rodenkirch(Feb. 2016) | status unknown | ||
near-square prime | A005574 | 1150678 | S. Scott (Mar. 12, 2008) | generalized Fermat prime | proven prime (http://primes.utm.edu/primes/page.php?id=84401) | |
near-square prime | A067201 | 110174 | D. Broadhurst (Apr. 8, 2006) | proven prime (http://primes.utm.edu/primes/page.php?id=77539) | ||
near-square prime ("Carol" prime) | A091515 | 653490 | 393441 | M. Rodenkirch (Jun. 15, 2016) | ? | status unknown |
near-square prime ("Kynea" prime) | A091513 | 661478 | 398250 | M. Rodenkirch (Jun. 19, 2016) | ? | status unknown |
NSW number | A113501 | 86121 | 65931 | E. W. Weisstein (Jan. 25, 2007) | subset of Pell-Lucas numbers | probable prime |
Padovan sequence number | A112882 | 727734 | 88874 | E. W. Weisstein (Apr. 7, 2011) | 729586 (E. W. Weisstein, Apr. 10, 2011) | status unknown |
palindromic prime | A002385 | 474501 | S. Batalov (Nov. 2014) | ? | proven prime (http://primes.utm.edu/top20/page.php?id=53#records) | |
partition function P | A046063 | 1000007396 | 35219 | E. W. Weisstein (Feb. 12, 2017) | 1000007862 (E. W. Weisstein, Feb. 12, 2017) | status unknown |
partition function Q | A035359 | 81191852 | ? | M. Alekseyev (Jul. 10, 2015) | (M. Alekseyev, Jul. 10, 2015) | status unknown |
Pell number | A096650 | 90197 | 34525 | T. D. Noe | 277096 (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 | A099088 | 172243 | 65931 | E. W. Weisstein (Jan. 25, 2007) | 270460 (E. W. Weisstein, Oct. 9, 2015) | probable prime; largest proven prime has index 9679 and 3705 digits (http://primes.utm.edu/primes/page.php?id=27783) |
Perrin sequence number | A112881 | 581132 | 70970 | E. W. Weisstein (Feb. 25, 2011) | 601463 (E. W. Weisstein, Oct. 9, 2015) | probable prime |
phi-prime | A064119 | 97241 | 97241 | E. W. Weisstein (Jun. 3, 2009) | 500000 (M. Rodenkirch, Jun. 20, 2017) | status unknown |
pi-prime | A060421 | 613373 | 613373 | A. Bondrescu (May 29, 2016) | ? | status unknown |
Pierpont prime | A005109 | 1529928 | A. Brady (Apr. 4, 2009) | ? | proven prime (http://primes.utm.edu/primes/page.php?id=87449) | |
power difference prime | A072164 | 7918 | 30870 | H. Lifchitz (2001) | probable prime | |
primefree sequence inverted primes | A108156 | 182220 | 38098 | E. W. Weisstein (Feb. 4, 2006) | 194202 (E. W. Weisstein, May 5, 2006) | status unknown |
primorial prime | A057704 | 85586 | 476310 | PrimeGrid (Mar. 5, 2012) | 154674 (PrimeGrid, Oct. 30, 2015) | proven prime |
primorial prime | A014545 | 33237 | 169966 | ? | 154670 (PrimeGrid, Oct. 30, 2015) | proven prime (http://primes.utm.edu/primes/page.php?id=21658) |
Pythagoras-constant prime | A115377 | 11540 | 11540 | E. W. Weisstein (Jan. 21, 2005) | 187336 (E. W. Weisstein, Feb. 8, 2016) | proven prime |
reflected pi-palindromic prime | A119351 | 2056 | 4111 | E. W. Weisstein (May 15, 2006) | 56755 (E. W. Weisstein, Mar. 21, 2009) | status unknown |
repunit | A004023 | 270343 | 270343 | M. Voznyy and A. Budnyy (Jul. 11, 2007) | probable prime; the largest proven prime has index 1031 (http://primes.utm.edu/top20/page.php?id=57) | |
Smarandache prime | - | - | - | 344869 (Great Smarandache PRPrime search, Dec. 5, 2016) | none known | |
Smarandache-Wellin prime | A046035 | 1429 | 5719 | ? | 1500000 (M. Rodenkirch, early 2016) | probable prime |
Soldner-constant prime | A122422 | 19474 | 19474 | E. W. Weisstein (Sep. 6, 2006) | 21757 (E. W. Weisstein, Sep. 7, 2006) | probable prime |
Sophie Germain prime | A005384 | 388342 | PrimeGrid (Feb. 29, 2016) | 2559254540925 (PrimeGrid, Feb. 29, 2016) | proven prime (http://primes.utm.edu/primes/page.php?id=121330) | |
super Catalan number | A092839 | 216 | 161 | E. W. Weisstein (Mar. 2004) | 584400 (E. W. Weisstein, Mar. 16, 2004) | proven prime (trivial) |
tau function | A135430 | 555339 | N. Lygeros and O. Rozier (Sep. 2015) | ? | probable prime | |
tetradic number | A068188 | 180055 | D. Bedwell (Sep. 8, 2009) | ? | proven prime (http://primes.utm.edu/top20/page.php?id=53#records) | |
tetranacci number | A104534 | 155182 | 44228 | E. W. Weisstein (Oct. 28, 2005) | 236965 (E. W. Weisstein, Mar. 29, 2009) | probable prime |
Thâbit ibn Kurrah prime | A002235 | 11895718 | 3580969 | PrimeGrid (Jun. 6, 2015) | 12078521 (PrimeGrid, Nov. 3, 2015) | proven prime (http://primes.utm.edu/primes/page.php?id=120038) |
Theodorus-constant prime | A119344 | 170657 | 170657 | E. W. Weisstein (Aug. 18, 2013) | 193924 (E. W. Weisstein, Feb. 8, 2016) | status unknown |
tribonacci number | A092835 | 96878 | 25639 | K. Duszenko (Aug. 2003) | 291217 (E. W. Weisstein, Mar. 21, 2009) | probable prime |
twin primes | A001359 | 388342 | PrimeGrid (Sep. 14, 2016) | ? | proven prime (http://primes.utm.edu/top20/page.php?id=1#records) | |
Wagstaff prime | A000978 | 13372531 | 4025533 | R. Propper (Sep. 2013) | ? | probable prime |
Wall-Sun-Sun prime | - | - | (PrimeGrid, Oct. 30, 2015) | none known | ||
Wieferich prime | A001220 | 3511 | 4 | (PrimeGrid, Oct. 30, 2015) | proven prime (trivial) | |
Wilson prime | A007540 | 563 | 3 | (Costa et al. 2012) | proven prime (trivial) | |
Wolstenholme number | A111354 | 48590 | 42146 | T. D. Noe (Mar. 8, 2006) | probable prime | |
Wolstenholme prime | A088164 | 2124679 | 7 | (McIntosh 2004) | proven prime (trivial) | |
Woodall prime | A002234 | 17016602 | 5122515 | D. Bertolotti (Mar. 22, 2018) | 16838832 (PrimeGrid, Mar. 22, 2018) | proven prime (http://primes.utm.edu/primes/page.php?id=124539) |
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REFERENCES:
--. "The Great Smarandache PRPrime search." http://smarandache.ddns.net:1200/server_stats.html.
Caldwell, C. and Gallot, Y. "On the Primality of and ." 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 . 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 PRPNet. "Factorial Prime Search - Primes by User." http://prpnet.primegrid.com:12002/user_primes.html.
PrimeGrid PRPNet. "Factorial Prime Search - Server Statistics." http://prpnet.primegrid.com:12002.
PrimeGrid PRPNet. "Primorial Prime Search - Primes by User." http://prpnet.primegrid.com:12008/user_primes.html.
PrimeGrid PRPNet. "Primorial Prime Search - Server Statistics." http://prpnet.primegrid.com:12008.
PrimeGrid PRPNet. "Wall-Sun-Sun Prime Search." http://prpnet.primegrid.com:13001.
PrimeGrid PRPNet. "Wieferich Prime Search." http://prpnet.primegrid.com:13000.
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)." 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."
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مخاطر خفية لمكون شائع في مشروبات الطاقة والمكملات الغذائية
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"آبل" تشغّل نظامها الجديد للذكاء الاصطناعي على أجهزتها
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المجمع العلميّ يُواصل عقد جلسات تعليميّة في فنون الإقراء لطلبة العلوم الدينيّة في النجف الأشرف
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