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A Chen prime is a prime number for which is either a prime or semiprime. Chen primes are named after Jing Run Chen who proved in 1966 that there are infinitely many such primes (Chen's theorem).The first Chen primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (OEIS A109611). The first primes that are not Chen primes are 43, 61, 73, 79, 97, 103, 151, ... (OEIS A102540).The lesser of any twin prime is always a Chen prime. Apart from twin prime records, the largest known Chen prime known as of Oct. 2005 was(https://primes.utm.edu/primes/page.php?id=75857),which has 70301 digits.There are infinitely many cases of 3 Chen primes in arithmetic progression (Green and Tao 2005). The following 3074-digit case produces Chen primes for , 1, 2, where denotes the primorial:

A palindromic prime is a number that is simultaneously palindromic and prime. The first few (base-10) palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, ... (OEIS A002385; Beiler 1964, p. 228). The number of palindromic primes less than a given number are illustrated in the plot above. The number of palindromic numbers having , 2, 3, ... digits are 4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, ... (OEIS A016115; De Geest) and the total number of palindromic primes less than 10, , , ... are 4, 5, 20, 20, 113, 113, 781, ... (OEIS A050251). Gupta (2009) has computed the numbers of palindromic primes up to .The following table lists palindromic primes in various small bases. OEISbase- palindromic primes2A11769711, 101, 111, 10001, 11111, 1001001, 1101011, ...3A1176982, 111, 212, 12121, 20102, 22122, ...4A1176992, 3, 11, 101, 131, 323, 10001, 11311, 12121, ...5A1177002, 3, 111, 131, 232, 313, 414, 10301, 12121,..

A sequence of primes is a Cunningham chain of the first kind (second kind) of length if () for , ..., . Cunningham primes of the first kind are Sophie Germain primes.It is conjectured there are arbitrarily long Cunningham chains. The longest known Cunningham chains are of length 17, with the first examples found corresponding to (first kind; J. Wroblewski, May 2008) and (second kind; J. Wroblewski, Jun. 2008).The smallest prime beginning a complete Cunningham chain of the first kind of lengths , 2, ... are 13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769, 26089808579, ... (OEIS A005602).The smallest prime beginning a complete Cunningham chain of the second kind of lengths , 2, ... are 11, 7, 2, 2131, 1531, 33301, 16651, 15514861, 857095381, 205528443121, ... (OEIS A005603)...

Pairs of primes of the form (, ) are called cousin primes. The first few are (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), ... (OEIS A023200 and A046132).A large pair of cousin (proven) primes start with(1)where is a primorial. These primes have 10154 digits and were found by T. Alm, M. Fleuren, and J. K. Andersen (Andersen 2005).As of Jan. 2006, the largest known pair of cousin (probable) primes are(2)which have 11311 digits and were found by D. Johnson in May 2004.According to the first Hardy-Littlewood conjecture, the cousin primes have the same asymptotic density as the twin primes,(3)(4)where (OEIS A114907) is the twin primes constant.An analogy to Brun's constant, the constant(5)(omitting the initial term ) can be defined. Using cousin primes up to , the value of is estimated as(6)..

Sexy primes are pairs of primes of the form (, ), so-named since "sex" is the Latin word for "six.". The first few sexy prime pairs are (5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), ... (OEIS A023201 and A046117). As of November 2005, the largest known sexy prime pair starts with(1)where is a primorial. These primes have 10154 digits and were found by M. Fleuren, T. Alm, and J. K. Andersen (Andersen 2005).Sexy constellations also exist. The first few sexy triplets (i.e., numbers such that each of is prime but is not prime) are (7, 13, 19), (17, 23, 29), (31, 37, 43), (47, 53, 59), ... (OEIS A046118, A046119, and A046120). As of October 2005, the largest known sexy triplet starts with(2)These primes have 5132 digit digits and were found by Davis (2005).The first few sexy quadruplets are (11, 17, 23, 29), (41, 47, 53, 59), (61, 67, 73, 79), (251, 257, 263, 269),..

A prime gap of length is a run of consecutive composite numbers between two successive primes. Therefore, the difference between two successive primes and bounding a prime gap of length is , where is the th prime number. Since the prime difference function(1)is always even (except for ), all primes gaps are also even. The notation is commonly used to denote the smallest prime corresponding to the start of a prime gap of length , i.e., such that is prime, , , ..., are all composite, and is prime (with the additional constraint that no smaller number satisfying these properties exists).The maximal prime gap is the length of the largest prime gap that begins with a prime less than some maximum value . For , 2, ..., is given by 4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ... (OEIS A053303).Arbitrarily large prime gaps exist. For example, for any , the numbers , , ..., are all composite (Havil 2003, p. 170). However, no general method..

An arithmetic progression of primes is a set of primes of the form for fixed and and consecutive , i.e., . For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression of primes with difference 210.It had long been conjectured that there exist arbitrarily long sequences of primes in arithmetic progression (Guy 1994). As early as 1770, Lagrange and Waring investigated how large the common difference of an arithmetic progression of primes must be. In 1923, Hardy and Littlewood (1923) made a very general conjecture known as the k-tuple conjecture about the distribution of prime constellations, which includes the hypothesis that there exist infinitely long prime arithmetic progressions as a special case. Important additional theoretical progress was subsequently made by van der Corput (1939), who proved than there are infinitely many triples of primes in arithmetic progression, and Heath-Brown (1981),..

Just as many interesting integer sequences can be defined and their properties studied, it is often of interest to additionally determine which of their elements are prime. The following table summarizes the indices of the largest known prime (or probable prime) members of a number of named sequences.sequenceOEISdigitsdiscoverersearch limitcommentsalternating factorialA00127259961260448M. Rodenkirch (Sep. 18, 2017)100000 (M. Rodenkirch, Dec. 15, 2017)finite sequence; largest certified prime has index 661; the rest are probable primesApéry-constant primeA119334141141E. W. Weisstein (May 14, 2006)9089 (E. W. Weisstein, Mar. 22, 2008)status unknownApéry number A092825662410136E. W. Weisstein (Mar. 2004) (E. W. Weisstein, Mar. 2004)probable primeApéry number 87E. W. Weisstein..

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