A093364 -id:A093364 - cp's OEIS frontend

A122763 A093364 with duplicates removed.

0, 1, 2, 6, 30, 150, 210, 13860, 60060, 420420, 4144140, 9699690, 87297210, 717777060, 4180566390, 18846497670, 26004868890

Offset: 1

Roger L. Bagula, Sep 22 2006

			Succeedingly longer arithmetic progressions of primes where the end value is smallest (A005115) start 2,2+j,3+2j,5+6j,5+6j,7+30j,7+150j... This sequence collects the unique values of the common differences.

Cf. A093364, A005115.

Edited by Ralf Stephan, May 22 2014

A006560 Smallest starting prime for n consecutive primes in arithmetic progression.

Original entry on oeis.org

2, 2, 3, 251, 9843019, 121174811

Offset: 1

N. J. A. Sloane

The primes following a(5) and a(6) occur at a(n)+30*k, k=0..(n-1). a(6) was found by Lander and Parkin. The next term requires a spacing >= 210. The expected size is a(7) > 10^21 (see link). - Hugo Pfoertner, Jun 25 2004
From Daniel Forgues, Jan 17 2011: (Start)
It is conjectured that there are arithmetic progressions of n consecutive primes for any n.
Common differences of first and smallest AP of n >= 1 consecutive primes: {0, 1, 2, 6, 30, 30, >= 210, >= 210, >= 210, >= 210, >= 2310, ...} (End)
a(7) <= 71137654873189893604531, found by P. Zimmermann, cf. J. K. Andersen link. - Bert Dobbelaere, Jul 27 2022

			First and smallest occurrence of n, n >= 1, consecutive primes in arithmetic progression:
a(1) = 2: (2) (degenerate arithmetic progression);
a(2) = 2: (2, 3) (degenerate arithmetic progression);
a(3) = 3: (3, 5, 7);
a(4) = 251: (251, 257, 263, 269);
a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139);
a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jens Kruse Andersen, The smallest known CPAP-k.
Thomas Bloom, Let k >= 3. Are there k consecutive primes in arithmetic progression?, Erdős Problems.
Chris K. Caldwell, Consecutive Primes in Arithmetic Progression
Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, and P. Zimmermann, Ten consecutive primes in arithmetic progression, Math. Comp., Vol. 71, No. 239 (2002) 1323-1328.
Daniel Forgues, Wiki about consecutive primes in arithmetic progression.
L. J. Lander and T. R. Parkin, Consecutive primes in arithmetic progression, Math. Comp., Vol. 21, No. 99 (1967) p 489.
Terence Tao, Erdős problem database, see no. 141.
Manfred Toplic, The nine and ten primes project, 2004.
Index entries for sequences related to primes in arithmetic progressions

Cf. A005115, A006562, A093364, A126989.
a(5) corresponds to A052243(20) followed by A052243(21) 9843049.
Cf. A089180: indices primes a(n).
Cf. A054800: start of 4 consecutive primes in arithmetic progression (CPAP-4), A033451: start of CPAP-4 with common difference 6, A052239: start of first CPAP-4 with common difference 6n.
Cf. A059044: start of 5 consecutive primes in arithmetic progression, A210727: CPAP-5 with common difference 60.
Cf. A058362: start of 6 consecutive primes in arithmetic progression.

Join[{2},Table[SelectFirst[Partition[Prime[Range[691*10^4]],n,1], Length[ Union[ Differences[ #]]] == 1&][[1]],{n,2,6}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 10 2019 *)

a(n) = A000040(A089180(n)), or A089180(n) = A000720(a(n)). - M. F. Hasler, Oct 27 2018

Edited by Daniel Forgues, Jan 17 2011

A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

Original entry on oeis.org

2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 249037, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111

Offset: 1

N. J. A. Sloane

In other words, smallest prime which is at the end of an arithmetic progression of n primes.
For the corresponding values of the first term and the common difference, see A113827 and A093364. For the actual arithmetic progressions, see A133277.
One may also minimize the common difference: this leads to A033189, A033188 and A113872.
One may also specify that the first term is the n-th prime and then minimize the common difference (or, equally, the last term): this leads to A088430 and A113834.
One may also ask for n consecutive primes in arithmetic progression: this gives A006560.

			n, AP, last term
1 2 2
2 2+j 3
3 3+2j 7
4 5+6j 23
5 5+6j 29
6 7+30j 157
7 7+150j 907
8 199+210j 1669
9 199+210j 1879
10 199+210j 2089
11 110437+13860j 249037
12 110437+13860j 262897
..........................
a(11)=249037 since 110437,124297,...,235177,249037 is an arithmetic progression of 11 primes ending with 249037 and it is the least number with this property.

R. K. Guy, Unsolved Problems in Number Theory, A5.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
Thomas Bloom, Problem 200 and Problem 215, Erdős Problems.
Harvey Dubner and Harry Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188 [math.NT], 2004-2007.
Ben Green and Terence Tao, A bound for progressions of length k in the primes
Andrew Granville, Prime number patterns, American Mathematical Monthly 115 (2008), pp. 279-296.
Paul A. Pritchard, Andrew Moran and Anthony Thyssen, Twenty-two primes in arithmetic progression, Math. Comp. 64 (1995), no. 211, 1337-1339.
Terence Tao, Erdős problem database, see no. 200, 215.
Index entries for sequences related to primes in arithmetic progressions

For the associated gaps, see A093364. For the initial terms, see A113827. For the arithmetic progressions, see A133277.

Cf. A006560, A096003, A113830-A113834, A088430.

(* This program will generate the 4 to 12 terms to use a[n_] to generate term 13 or higher, it will have a prolonged run time. *) a[n_] := Module[{i, p, found, j, df, k}, i = 1; While[i++; p = Prime[i]; found = 0; j = 0; While[j++; df = 6*j; (p > ((n - 1)*df)) && (found == 0), found = 1; Do[If[! PrimeQ[p - k*df], found = 0], {k, 1, n - 1}]]; found == 0]; p]; Table[a[i], {i, 4, 12}]

Green & Tao prove that this sequence is infinite, and further a(n) < 2^2^2^2^2^2^2^2^O(n). Granville conjectures that a(n) <= n! + 1 for n >= 3 and give a heuristic suggesting a(n) is around (exp(1-gamma) n/2)^(n/2). - Charles R Greathouse IV, Feb 26 2013

a(11)-a(13) from Michael Somos, Mar 14 2004
a(14) and corrected version of a(7) from Hugo Pfoertner, Apr 27 2004
a(15)-a(17) from Don Reble, Apr 27 2004
a(18)-a(21) from Granville's paper, Jan 26 2006
Entry revised by N. J. A. Sloane, Jan 26 2006, Oct 17 2007

A133277 Triangle read by rows: row n gives the arithmetic progression of n primes with minimal final term, cf. A005115.

Original entry on oeis.org

2, 2, 3, 3, 5, 7, 5, 11, 17, 23, 5, 11, 17, 23, 29, 7, 37, 67, 97, 127, 157, 7, 157, 307, 457, 607, 757, 907, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089, 110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037

Offset: 1

N. J. A. Sloane, Oct 17 2007

The first 10 rows (i.e., 55 terms) are the same as for A133276 (where the common distance is minimal), but here T(11,1) = a(56) = 110437 while A133276(11,1) = 60858179. - M. F. Hasler, Jan 02 2020

For any prime p there is a p-AP (arithmetic progression of p primes) starting with p, where the common distance is given by A088430. For n between prime(k-1) and prime(k), there may be an n-AP starting at prime(k) (but not earlier) with a smaller common distance, given in A061558. - M. F. Hasler, Sep 17 2024

			Triangle begins:
    2;
    2,   3;
    3,   5,   7;
    5,  11,  17,  23;
    5,  11,  17,  23,   29;
    7,  37,  67,  97,  127,  157;
    7, 157, 307, 457,  607,  757,  907;
  199, 409, 619, 829, 1039, 1249, 1459, 1669;
  199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879;
  199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089;
  ...

For common differences, see A093364. For initial terms, see A113827. For final terms, see A005115.
Differs from A133276 (from T(11,1) = a(56) on).
See also A061558 (distance in earliest n-AP), A088430 (same for primes), A231017 (second term in p-AP starting with p), A061558 (distance of n-AP starting at the smallest possible prime).

A-numbers in the Name and Crossrefs sections corrected by Bobby Jacobs, Dec 10 2016

Name edited by M. F. Hasler, Jan 02 2020

A097824 Gaps associated with the arithmetic progressions of semiprimes in A096003.

Original entry on oeis.org

12, 12, 4, 52, 52, 72, 96, 198, 198, 114, 594, 48, 354, 1860, 3942, 2574, 2574, 2574, 20910, 20910, 9600, 9600, 152250, 152250, 152250, 152250, 1887270, 4667040, 4094790

Offset: 4

Hugo Pfoertner, Aug 27 2004

The terms a(1),a(2) and a(3) are omitted to avoid the ambiguity caused by the two progressions of length 3 ending at A096003(3)=14: a(3)=5 for (4,9,14) or a(3)=4 for (6,10,14).

			a(6)=4 because the 6 semiprimes in the progression ending at A096003(6)=221 are separated by an increment of 4: 201=3*67, 205=5*41, 209=11*19, 213=3*71, 217=7*31, 221=13*17.

Cf. A096003, A093364 gaps in arithmetic progressions of primes, A082919 clusters of 8 consecutive semiprimes.

a(26)-a(30) from Hugo Pfoertner, Sep 07 2004

a(31)-a(32) from Donovan Johnson, Jun 03 2012

A273921 Number of 11-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 15, 253, 5561

Offset: 1

Jean-Marc Rebert, Jun 04 2016

			The least 11-tuple is {110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037} (this is also the beginning of the least 12-tuple). This is one of the 15 11-tuples corresponding to a(6)=15.

See A113827.

Cf. A113827 (prime beginning minimal n-tuple of primes in AP).

Cf. A005115, A093364, A115608, A115609, A115610, A115611, A115612, A115613, A273919, A273920, A273922, A273923.

A273922 Number of 12-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 42, 715

Offset: 1

Jean-Marc Rebert, Jun 04 2016

			The least 12-tuple is {110437, 124297, 138157, 152017, 165877, 179737, 193597, 207457, 221317, 235177, 249037, 262897}.

See A113827.

Cf. A113827 (prime beginning minimal n-tuple of primes in AP).

Cf. A005115, A093364, A115608, A115609, A115610, A115611, A115612, A115613, A273919, A273920, A273921, A273923.

A273923 Number of 13-tuples of primes in arithmetic progression less than 10^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 52

Offset: 1

Jean-Marc Rebert, Jun 04 2016

			The least 13-tuple is {4943, 65003, 125063, 185123, 245183, 305243, 365303, 425363, 485423, 545483, 605543, 665603, 725663}.

See A113827.

Cf. A113827 (prime beginning minimal n-tuple of primes in AP).

Cf. A005115, A093364, A115608, A115609, A115610, A115611, A115612, A115613, A273919, A273920, A273921, A273922.

A354376 Smallest prime which is at the end of an arithmetic progression of exactly n primes.

Original entry on oeis.org

2, 3, 7, 43, 29, 157, 907, 2351, 5179, 2089, 375607, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111

Offset: 1

Bernard Schott, May 24 2022

Equivalently: Let i, i+d, i+2d, ..., i+(n-1)d be an arithmetic progression of exactly n primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.
The word "exactly" requires both i-d and i+n*d to be nonprime; without "exactly", we get A005115.
For the corresponding values of the first term, and the common difference, see A354377 and A354484. For the actual arithmetic progressions, see A354485.
The primes in these arithmetic progressions need not be consecutive. (The smallest prime at the start of a run of exactly n consecutive primes in arithmetic progression is A006560(n).)
a(n) != A005115(n), because A005115(n) + A093364(n) is prime for n = 4, 8, 9, 11. - Michael S. Branicky, May 24 2022

			The arithmetic progression (5, 11, 17, 23) with common difference 6 contains 4 primes, but 29 = 23+6 is also prime, hence a(4) != 23.
The arithmetic progression (7, 19, 31, 43) with common difference 12 also contains 4 primes, and 7-12 < 0 and 43+12 = 55 is composite; moreover this arithmetic progression is the smallest such progression with exactly 4 primes, hence a(4) = 43.

R. K. Guy, Unsolved Problems in Number Theory, A5, Arithmetic progressions of primes.

Cf. A005115, A006560, A093364, A354377, A354484, A354485.

from sympy import isprime, nextprime
def a(n):
    if n < 3: return [2, 3][n-1]
    p = 2
    while True:
        for d in range(2, (p-3)//(n-1)+1, 2):
            if isprime(p+d) or isprime(p-n*d): continue
            if all(isprime(p-j*d) for j in range(1, n)): return p
        p = nextprime(p)
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, May 24 2022

a(4) corrected and a(8)-a(13) from Michael S. Branicky, May 24 2022

a(14)-a(21) derived using A005115 and A093364 by Michael S. Branicky, May 24 2022

A274825 Gaps associated with the arithmetic progressions in A278735.

Original entry on oeis.org

0, 2, 6, 78, 8640, 1155000, 82419540

Offset: 1

Bobby Jacobs, Dec 08 2016

The first set of 4 prime-indexed primes in arithmetic progression (353, 431, 509, and 587) contains consecutive terms of A142160.

			a(4) = 78 because 353 = prime(prime(20)), 431 = prime(prime(23)), 509 = prime(prime(25)), 587 = prime(prime(28)), and 431-353 = 509-431 = 587-509 = 78.
For the corresponding arithmetic progressions, see _Charles R Greathouse IV_'s example in A278735. - _Bobby Jacobs_, Jan 02 2017

Cf. A006450, A093364, A142160, A278735, A279021, A279062.

a(7) from Bobby Jacobs, Jan 02 2017

cp's OEIS Frontend

A122763 A093364 with duplicates removed.

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A006560 Smallest starting prime for n consecutive primes in arithmetic progression.

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A005115 Let i, i+d, i+2d, ..., i+(n-1)d be an n-term arithmetic progression of primes; choose the one which minimizes the last term; then a(n) = last term i+(n-1)d.

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A133277 Triangle read by rows: row n gives the arithmetic progression of n primes with minimal final term, cf. A005115.

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A097824 Gaps associated with the arithmetic progressions of semiprimes in A096003.

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A273921 Number of 11-tuples of primes in arithmetic progression less than 10^n.

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A273922 Number of 12-tuples of primes in arithmetic progression less than 10^n.

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A273923 Number of 13-tuples of primes in arithmetic progression less than 10^n.

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A354376 Smallest prime which is at the end of an arithmetic progression of exactly n primes.

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