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 | 110000 (S. Plouffe, Apr. 13, 2024) | 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  |