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.

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)

Here,

			(1)
			(2)
			(3)

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."