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 ${\displaystyle a_{n}=n2^{n}+1.}$
• Factorial prime – a prime that appears in either of the sequences ${\displaystyle a_{n}=n!-1}$ or ${\displaystyle b_{n}=n!+1.}$
• Fermat prime – a prime that appears in the sequence of Fermat numbers ${\displaystyle a_{n}=2^{2^{n))+1.}$
• 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 ${\displaystyle a_{n}=2^{n}-1.}$
• Primorial prime – a prime that appears in either of the sequences ${\displaystyle a_{n}=n\#-1}$ or ${\displaystyle b_{n}=n\#+1.}$
• Pythagorean prime – a prime that appears in the sequence ${\displaystyle a_{n}=4n+1.}$
• Woodall prime – a prime that appears in the sequence of Woodall numbers ${\displaystyle a_{n}=n2^{n}-1.}$

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.

## References

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