In mathematics, an integer sequence prime is a prime number found as a member of an integer sequence. For example, the 8th Delannoy number, 265729, is prime. A challenge in empirical mathematics is to identify large prime values in rapidly growing sequences.

A common subclass of integer sequence primes are constant primes, formed by taking a constant real number and considering prefixes of its decimal representation, omitting the decimal point. For example, the first 6 decimal digits of the constant π, approximately 3.14159265, form the prime number 314159, which is therefore known as a pi-prime (sequence A005042 in the OEIS). Similarly, a constant prime based on e is called an e-prime.

Other examples of integer sequence primes include:

• Cullen prime – a prime that appears in the sequence of Cullen numbers $a_{n}=n2^{n}+1.$ • Factorial prime – a prime that appears in either of the sequences $a_{n}=n!-1$ or $b_{n}=n!+1.$ • Fermat prime – a prime that appears in the sequence of Fermat numbers $a_{n}=2^{2^{n))+1.$ • Fibonacci prime – a prime that appears in the sequence of Fibonacci numbers.
• Lucas prime – a prime that appears in the Lucas numbers.
• Mersenne prime – a prime that appears in the sequence of Mersenne numbers $a_{n}=2^{n}-1.$ • Primorial prime – a prime that appears in either of the sequences $a_{n}=n\#-1$ or $b_{n}=n\#+1.$ • Pythagorean prime – a prime that appears in the sequence $a_{n}=4n+1.$ • Woodall prime – a prime that appears in the sequence of Woodall numbers $a_{n}=n2^{n}-1.$ The On-Line Encyclopedia of Integer Sequences includes many sequences corresponding to the prime subsequences of well-known sequences, for example A001605 for Fibonacci numbers that are prime.

• Weisstein, Eric W. "Integer Sequence Primes". MathWorld.