A123556 -id:A123556 - cp's OEIS frontend

A359410 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 6 elements.

30, 60, 90, 120, 180, 240, 270, 300, 330, 360, 390, 450, 480, 510, 540, 570, 600, 660, 690, 720, 750, 780, 810, 870, 900, 930, 960, 990, 1020, 1080, 1110, 1140, 1170, 1200, 1230, 1290, 1320, 1350, 1380, 1410, 1440, 1500, 1530, 1560, 1590, 1620, 1650, 1710, 1740

Offset: 1

Bernard Schott, Jan 29 2023

nonn

The 6 elements are not necessarily consecutive primes.
A342309(d) gives the first element of the smallest AP with 6 elements whose common difference is a(n) = d.
All the terms are positive multiples of 30 (A249674) but are not multiples of 7 and also must not belong to A206041; indeed, terms d' in A206041 correspond to the longest possible APs of primes that have exactly 7 elements with this common difference d'.

			d = 30 is a term because the longest possible APs of primes with common difference d = 30 all have 6 elements; the first such APs start with 7, 107, 359, .... The smallest one is (7, 37, 67, 97, 127, 157); then 187 = 11*17.
d = 60 is another term because the longest possible APs of primes with common difference d = 60 all have 6 elements; the first such APs start with 11, 53, 641, .... The smallest one is (11, 71, 131, 191, 251, 311); then 371 = 7*53.
d = 150 is not a term because the longest possible AP of primes with common difference d = 150 is (7, 157, 307, 457, 607, 757, 907) which has 7 elements; this last one is unique.

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, A342309.
Subsequence of A249674.
Longest AP of prime numbers with exactly k elements: A007921 (k=1), A359408 (k=2), A206037 (k=3), A359409 (k=4), A206039 (k=5), this sequence (k=6), A206041 (k=7), no sequence for (k=8) and (k=9), A360146 (k=10), A206045 (k=11).

filter := d -> (irem(d, 30) = 0) and (irem(d, 7) <> 0) and not (isprime(7+d) and isprime(7+2*d) and isprime(7+3*d) and isprime(7+4*d) and isprime(7+5*d) and isprime(7+6*d)): select(filter, [$(1 .. 1740)]);

m is a term iff A123556(m) = 6.

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

Original entry on oeis.org

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.

A124064 Table read by rows: T(d,k) (d >= 1, k >= 1) = smallest prime p of k (not necessarily consecutive) primes in arithmetic progression with common difference d.

Original entry on oeis.org

2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 5, 5, 5, 5, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 5, 5, 5, 5, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 2, 5, 5, 5, 2, 2, 3, 3, 2, 2, 2, 7, 2, 2, 5, 5, 59, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 2, 7, 7, 7, 7, 7, 2, 2, 5, 2, 2, 3, 3, 2, 2, 2, 5, 7, 31, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 5, 5, 5, 5

Offset: 1

R. J. Mathar, Nov 04 2006

			Table begins:
d \k|..1..2..3..4..5..
----+-----------------
..1.|..2..2
..2.|..2..3..3
..3.|..2..2
..4.|..2..3..3
..5.|..2..2
..6.|..2..5..5..5..5
..7.|..2
..8.|..2..3..3
..9.|..2..2
.10.|..2..3..3
.11.|..2..2
.12.|..2..5..5..5..5
.13.|..2
.14.|..2..3..3
.15.|..2..2
.16.|..2..3
.17.|..2..2
.18.|..2..5..5..5
.19.|..2
.20.|..2..3..3
T(24,4) = 59 since (59,83,107,131) is the first A.P. of 4 primes with difference 24.

R. J. Mathar, Table for d <= 1000 (PDF)

Cf. A087242 (column k=2), A124570 (semiprimes analog), A249207.

Assuming the k-tuples conjecture, A123556 gives lengths of table rows.

T(n,1) = 2.

lim n->inf (a(n)/n) = SUM(p prime; (p-1)/(#(p-1)) = 2.92005097731613471209+

Edited by David W. Wilson, Nov 05 2006 and Nov 25 2006

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.

A125025 Lengths of rows in A124570.

Original entry on oeis.org

3, 8, 3, 8, 3

Offset: 1

Jonathan Vos Post, Nov 15 2006

This sequence is to A124570 as A123556 is to A124064.
a(n) is at most A053669(n)^2, with equality if and only if A053669(n)^2 is the first semiprime in the corresponding arithmetic progression. - Charlie Neder, Jan 10 2019
By subsampling a given arithmetic progression with k terms and distance d one may generate an arithmetic progression of a larger distance d*b, b=1,2,3...., with 1+(k-1)/b terms: a(b*n) >= 1+floor( (a(n)-1)/b ), b=1,2,3..... - R. J. Mathar, Aug 02 2021
18 <= a(6) <= 24. - Jinyuan Wang, Aug 06 2021
a(12) >= 17.  a(18) >= 18.  a(24) >= 18.  a(30) >= 21.  a(36) >= 19.  a(42) >= 20. a(6006) >= 24 (starting 652744562555081).  Hugo van der Sanden, Aug 14 2021

Cf. A001358, A123556, A124064, A124570, A091016 (for n=6).

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

cp's OEIS Frontend

A359410 Integers d such that the longest possible arithmetic progression (AP) of primes with common difference d has exactly 6 elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A124064 Table read by rows: T(d,k) (d >= 1, k >= 1) = smallest prime p of k (not necessarily consecutive) primes in arithmetic progression with common difference d.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A125025 Lengths of rows in A124570.

Views

Author

Keywords

Comments

Crossrefs

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

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

PARI

Extensions