Integer sequence prime

Last updated January 25, 2023

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 $a_{n}=n2^{n}+1.$
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.

Related Research Articles

In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form $M n = 2 n - 1$ for some integer $n$ . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If $n$ is a composite number then so is $2 n - 1$ . Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form $M p = 2 p - 1$ for some prime $p$ .

In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form

A palindromic number is a number that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term palindromic is derived from palindrome, which refers to a word whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers are:

In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system. The word is a portmanteau of repeated and digit. Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes.

In mathematics, a Cullen number is a member of the integer sequence $. Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers.$

In number theory, a Woodall number (W_n) is any natural number of the form

In mathematics, a k-hyperperfect number is a natural number n for which the equality n = 1 + k(σ − n − 1) holds, where σ(n) is the divisor function. A hyperperfect number is a k-hyperperfect number for some integer k. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect.

In number theory, a Wieferich prime is a prime number p such that p² divides $2 p - 1 - 1$ , therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides $2 p - 1 - 1$ . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.

In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.

The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.

In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin.

In mathematics, a double Mersenne number is a Mersenne number of the form

In number theory, a Wagstaff prime is a prime number of the form

A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.

Lucas pseudoprimes and Fibonacci pseudoprimes are composite integers that pass certain tests which all primes and very few composite numbers pass: in this case, criteria relative to some Lucas sequence.

In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple.

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol, or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as $e$ and $π$ occurring in such diverse contexts as geometry, number theory, statistics, and calculus.

A Proth number is a natural number N of the form $where k and n are positive integers, k is odd and . A Proth prime is a Proth number that is prime. They are named after the French mathematician François Proth. The first few Proth primes are$

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 .

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v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers