A231017 -id:A231017 - cp's OEIS frontend

A088430 a(n) = the least positive d such that for p=prime(n), the numbers p+0d, p+1d, p+2d, ..., p+(p-1)d are all primes.

1, 2, 6, 150, 1536160080, 9918821194590, 341976204789992332560, 2166703103992332274919550

Offset: 1

Zak Seidov, Sep 30 2003

Problem discussed by Russell E. Rierson: starting with given p, find the least d such that the arithmetic progression p,p+d,p+2d,... contains only primes. Obviously, the maximum number of prime terms is p and to reach that maximum, d must be a multiple of all smaller primes. For example, a(5) is a multiple of 2*3*5*7.

There can be other maximum-length prime progressions starting at p, with larger d. (Zak Seidov found d=4911773580 for p=11.)

			n AP Last term
--------------
1 2+i 3
2 3+2*i 7
3 5+6*i 29
4 7+150*i 907
5 11+1536160080*i 15361600811
6 13+9918821194590*i 119025854335093
7 17+341976204789992332560*i 5471619276639877320977
8 19+2166703103992332274919550*i 39000655871861980948551919

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 139-140.

Jens Kruse Andersen, Smallest AP-k with minimal start
Phil Carmody, a(7), NMBRTHRY Nov 2001
Andrew Granville, Prime number patterns
Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, arXiv:math/0404188 [math.NT], 2004-2007. [Background]
Paulo Ribenboim, Les records des nombres premiers, Séminaire de Philosophie et Mathématiques (8) (1987) 1-25.
Paulo Ribenboim, Prime number records, Coll. Math. J. 25 (4) (1994) 280-290.
Paulo Ribenboim, Euler's Famous prime generating polynomial and the class number of imaginary quadratic fields, (2000) p 91-111
Russell E. Rierson, Question About Prime Numbers.
Zak Seidov, Question About Prime Numbers.
Zak Seidov and others, Russell E. Rierson's Question About Prime Numbers, digest of 5 messages in primenumbers Yahoo group, Sep 29 - Oct 1, 2003.

See A113834 for last term in the progression, and A231017 for the 2nd term.

Cf. A061558, A231018, A002110.

A088430[n_] := Module[{p, m, d},
   p = Prime[n]; m = Product[Prime[i], {i, 1, n - 1}];
   d = m;
   While[! AllTrue[Table[p + i*d, {i, 1, p - 1}], PrimeQ], d = d + m];
   Return[d];
   ];
Table[A088430[n], {n, 1, 8}] (* Robert Price, Mar 27 2019 *)

a(n) = A231017(n) - prime(n). - Jonathan Sondow, Nov 08 2013
a(n) = A061558(prime(n)). - Jens Kruse Andersen, Jun 30 2014
a(n) = A002110(n-1) * A231018(n). - Jeppe Stig Nielsen, Mar 16 2016

Edited by Don Reble, Oct 04 2003
a(7) was found by Phil Carmody. - Don Reble, Nov 23 2003
Entry revised by N. J. A. Sloane, Jan 25 2006
a(8) found by Wojciech Izykowski. - Jens Kruse Andersen, Jun 30 2014

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

A231406 Table read by rows, with n-th row the smallest non-constant p-term arithmetic progression of primes beginning with p = prime(n).

Original entry on oeis.org

2, 3, 3, 5, 7, 5, 11, 17, 23, 29, 7, 157, 307, 457, 607, 757, 907, 11, 1536160091, 3072320171, 4608480251, 6144640331, 7680800411, 9216960491, 10753120571, 12289280651, 13825440731, 15361600811, 13, 9918821194603, 19837642389193, 29756463583783, 39675284778373, 49594105972963, 59512927167553, 69431748362143, 79350569556733, 89269390751323, 99188211945913, 109107033140503, 119025854335093, 17, 341976204789992332577

Offset: 1

Jonathan Sondow, Nov 08 2013

For comments, references, and links, see the crossrefs.

			Triangle begins:
2 3
3 5 7
5 11 17 23 29
7 157 307 457 607 757 907
11 1536160091 3072320171 4608480251 6144640331 7680800411 9216960491 10753120571 12289280651 13825440731 15361600811

Jens Kruse Andersen, Table of n, a(n) for n = 1..77
Index entries for sequences related to primes in arithmetic progressions

For common differences see A088430, for initial terms see A000040, for 2nd terms see A231017, for last terms see A113834.

T(n,k) = prime(n) + k*A088430(n), 0 <= k < prime(n), n > 0.

A231018 a(n) = d(n)/p(n-1)# where d(n) > 0 is the common difference of the smallest p-term arithmetic progression of primes beginning with p = p(n) = n-th prime.

Original entry on oeis.org

1, 1, 1, 5, 7315048, 4293861989, 11387819007325752, 4244193265542951705

Offset: 1

Jonathan Sondow, Nov 08 2013

d(n) is the least d > 0 such that p, p+d, p+2d, ..., p+(p-1)d are all prime with p = p(n), and p(n-1)# = A002110(n-1) is a primorial.
d(n) is always a multiple of p(n-1)#.
a(5) and a(6) are due to G. Loh in 1986, and a(7) to Phil Carmody in 2001.
See A088430 and A231017 for more comments, references, links, and examples.

			Prime(3) = 5 and 5, 11, 17, 23, 29 is the smallest 5-term AP beginning with 5, so a(3) = (11-5)/p(2)# = 6/2*3 = 1.

Index entries for sequences related to primes in arithmetic progressions

Cf. A002110, A088430, A231017.

a(n) = A088430(n) / A002110(n) = (A231017(n) - prime(n)) / A002110(n).

a(8) due to Wojciech Izykowski in 2014 added by Jonathan Sondow, Aug 08 2019

cp's OEIS Frontend

A088430 a(n) = the least positive d such that for p=prime(n), the numbers p+0d, p+1d, p+2d, ..., p+(p-1)d are all primes.

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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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A231406 Table read by rows, with n-th row the smallest non-constant p-term arithmetic progression of primes beginning with p = prime(n).

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A231018 a(n) = d(n)/p(n-1)# where d(n) > 0 is the common difference of the smallest p-term arithmetic progression of primes beginning with p = p(n) = n-th prime.

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