Integer sequence prime

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. 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.

References

• Weisstein, Eric W., "Integer Sequence Primes", MathWorld.
• Weisstein, Eric W., "Constant Primes", MathWorld.
• Weisstein, Eric W., "Pi-Prime", MathWorld.
• Weisstein, Eric W., "e-Prime", MathWorld.