# Integer sequence prime

Short description: Prime number found as a member of an integer sequence

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 Euler's number, 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.