A342309 -id:A342309 - cp's OEIS frontend

A360146 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 10 elements.

210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, 2520, 2730, 2940, 3150, 3360, 3570, 3780, 3990, 4200, 4410, 4830, 5040, 5250, 5460, 5670, 5880, 6090, 6300, 6510, 6720, 7140, 7350, 7560, 7770, 7980, 8190, 8400, 8610, 8820, 9030, 9450, 9660, 9870, 10080, 10290, 10500, 10710, 10920

Offset: 1

Bernard Schott, Mar 09 2023

nonn

The 10 elements are not necessarily consecutive primes.
All the terms are positive multiples of 210 = 7# but are not multiples of 11 and also must not belong to A206045, where the first term is 1536160080; indeed, terms d' in A206045 correspond to the longest possible APs of primes that have exactly 11 elements with these common differences d'.
A342309(d) gives the first element of the smallest AP with 10 elements whose common difference is a(n) = d.

			d = 210 is a term because the longest possible APs of primes with common difference d = 210 all have 10 elements. The first such AP is (199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089), then 2299 = 11*209.
d = 420 is another term because the longest possible APs of primes with common difference d = 420 all have 10 elements; the first such APs start with 52879, 3544939, ... The smallest one is (52879, 53299, 53719, 54139, 54559, 54979, 55399, 55819, 56239, 56659), then 57079 = 11*5189.

Diophante, A1880. NP en PA (prime numbers in arithmetic progression) (in French).
Wikipedia, Primes in arithmetic progression.
Index entries for sequences related to primes in arithmetic progressions.

Cf. A123556, A173919, A342309.

Common differences for longest possible APs of primes with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), A359410 (k=6), A206041 (k=7), this sequence (k=10), A206045 (k=11).

A053669(n) = forprime(p=2, , if(n%p, return(p)));
f(n) = my(p=A053669(n)); for (i=1, p-1, if (!isprime(p+i*n), return(p-1))); p; \\ A123556
isok(n) = f(n) == 10; \\ Michel Marcus, Mar 10 2023

m is a term iff A123556(m) = 10.

A360735 Even integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

Original entry on oeis.org

16, 22, 26, 32, 44, 46, 52, 56, 58, 62, 70, 74, 76, 82, 86, 88, 92, 100, 106, 112, 116, 118, 122, 128, 130, 136, 140, 142, 146, 148, 152, 158, 160, 166, 170, 172, 176, 182, 184, 194, 196, 200, 202, 206, 212, 214, 218, 224, 226, 232, 236, 242, 244, 250, 254, 256, 262, 266, 268

Offset: 1

Bernard Schott, Feb 19 2023

nonn

Similar sequence with odd integers d is A040976 \ {0}.
Terms are even numbers that are not divisible by 3 and that are not also in A206037.
These longest corresponding APs are of the form (q, q+d) with q odd primes (see examples).
This subsequence of A359408 corresponds to the second case '2 is one less than prime 3' (see A173919); the first case is linked to A040976.
A342309(d) gives the first element of the smallest such AP with 2 elements whose common difference is a(n) = d.

			d = 16 is a term because the first longest APs of primes with common difference 16 are (3, 19), (7,23), (13, 29), ... and all have 2 elements because next elements should be respectively 35, 39 and 45 that are all composite; the first such AP that starts with A342309(16) = 3 is (3, 19).
d = 22 is a term because the first longest APs of primes with common difference 22 are (7, 29), (19, 41), (31, 53), ... and all have 2 elements because next elements should be respectively 51, 63 and 75 that are all composite; the first such AP that starts with A342309(22) = 7 is (7, 29).

Equals A359408 \ A040976.

Cf. A047235, A123556, A206037, A342309.

filter := d -> (irem(d, 2) = 0) and (irem(d, 3) <> 0) and not isprime(3+d) or isprime(3+d) and not isprime(3+2*d) : select(filter, [`$`(1 .. 270)]);
isA360735 := d -> isA047235(d) and not isA206037(d): # Peter Luschny, Mar 03 2023

Select[Range[2, 270, 2], Mod[#, 3] > 0 && Nand @@ PrimeQ[{# + 3, 2*# + 3}] &] (* Amiram Eldar, Mar 03 2023 *)

isok(d) = !(d%2) && (d%3) && !(isprime(d+3) && isprime(2*d+3)); \\ Michel Marcus, Mar 03 2023

If m is a term then A123556(m) = 2, but the converse is false: a counterexample is A123556(11) = 2 and 11 is not a term.

A284708 Smallest initial prime p for at least n primes in increasing arithmetic progression with a common difference less than p.

Original entry on oeis.org

2, 2, 3, 11, 37, 107, 409, 409, 409, 25471, 53173, 65003, 766439, 11797483

Offset: 1

Arkadiusz Wesolowski, Jan 09 2018

Conjecture: a(n) > A034386(n) for every n >= 4.
From Bernard Schott, Mar 15 2023: (Start)
Corresponding common differences are in A361492.
a(22) = 11410337850553 since it is the smallest term in a sequence of 22 primes in arithmetic progression, and the corresponding common difference 4609098694200 is < a(22) (see Penguin reference). (End)

			Smallest initial prime p, primes in arithmetic progression:
a(1) = 2: (2);
a(2) = 2: (2, 3);
a(3) = 3: (3, 5, 7);
a(4) = 11: (11, 17, 23, 29);
a(5) = 37: (37, 67, 97, 127, 157);
a(6) = 107: (107, 137, 167, 197, 227, 257);
a(7) = 409: (409, 619, 829, 1039, 1249, 1459, 1669);
a(8) = 409: (409, 619, 829, 1039, 1249, 1459, 1669, 1879);
a(9) = 409: (409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089);

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, entry 11410337850553, page 191.

Cf. A361492.

Cf. A123556, A342309.

isokd(p, n, d) = for (i=1, n, if (!isprime(p+(i-1)*d), return(0))); 1;
isokp(p, n) = for (d=1, p-1, if (isokd(p, n, d), return(1)););
a(n) = my(p=2); while (!isokp(p, n), p=nextprime(p+1)); p; \\ Michel Marcus, Mar 15 2023

Name edited by Bernard Schott, Mar 15 2023

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A360146 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 10 elements.

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A360735 Even integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has only two elements.

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A284708 Smallest initial prime p for at least n primes in increasing arithmetic progression with a common difference less than p.

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