A088430 -id:A088430 - cp's OEIS frontend

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.

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

A113832 Triangle read by rows: row n (n>=2) gives a set of n primes with the property that the pairwise averages are all primes, having the smallest largest element.

Original entry on oeis.org

3, 7, 3, 7, 19, 3, 11, 23, 71, 5, 29, 53, 89, 113, 3, 11, 83, 131, 251, 383, 5, 29, 113, 269, 353, 449, 509, 5, 17, 41, 101, 257, 521, 761, 881, 23, 431, 503, 683, 863, 1091, 1523, 1871, 2963, 31, 1123, 1471, 1723, 3463, 3571, 4651, 5563, 5743, 6991

Offset: 2

N. J. A. Sloane, Jan 25 2006

If there is more than one set with the same smallest last element, choose the lexicographically earliest solution.

For distinct primes, the solution for n=5 is {5, 29, 53, 89, 173}.

			Triangle begins:
3, 7
3, 7, 19
3, 11, 23, 71
5, 29, 53, 89, 113
3, 11, 83, 131, 251, 383
5, 29, 113, 269, 353, 449, 509
The set of primes generated by {5, 29, 53, 89, 113} is {17, 29, 41, 47, 59, 59, 71, 71, 83, 101}.

Antal Balog, The prime k-tuplets conjecture on average, in "Analytic Number Theory" (eds. B. C. Berndt et al.) Birkhäuser, Boston, 1990, pp. 165-204. [Background]

Toshitaka Suzuki, Table of n, a(n) for n = 2..91
Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
Andrew Granville, Prime number patterns

Cf. A113827-A113831, A113833, A113834, A088430.

See A115631 for the case when all pairwise averages are distinct primes.

More terms from T. D. Noe, Feb 01 2006

A113834 Last term of smallest p-term arithmetic progression of primes beginning with p, where p = n-th prime.

Original entry on oeis.org

3, 7, 29, 907, 15361600811, 119025854335093, 5471619276639877320977, 39000655871861980948551919

Offset: 1

N. J. A. Sloane, Jan 25 2006

See A088430 for further information and references.

			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

Based on the information in A088430.

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

A231017 Least prime q > p = prime(n) such that if d = q-p, then p, p+d, p+2d, ..., p+(p-1)d are all primes.

Original entry on oeis.org

3, 5, 11, 157, 1536160091, 9918821194603, 341976204789992332577, 2166703103992332274919569

Offset: 1

Jonathan Sondow, Nov 08 2013

Second term in the smallest non-constant p-term arithmetic progression (AP) of primes beginning with p = prime(n).
For any non-constant AP beginning with a prime p and ending with a prime, the maximum possible length is p, since p+pd is not prime for d > 0. If all the terms are prime, then the common difference d must be a multiple of all primes < p.
Ribenboim says that in 1986 G. Loh [Loeh] discovered a(5) and a(6), and that a(n) should exist for all n, but "in my opinion, this is so difficult that no one will prove [it], and no one will find a counterexample in the near future."
Phil Carmody found a(7) in 2001.
See the crossrefs for more comments, references, and links.

			Prime(3) = 5 and 5, 11, 17, 23, 29 is the smallest 5-term AP beginning with 5, so a(3) = 11.

P. Ribenboim, My Numbers, My Friends, Springer, 2000; p. 67.
P. Ribenboim, The Book of Prime Number Records, 2nd ed., Springer, 1989; p. 225.

Phil Carmody, a(7), NMBRTHRY November 2001.
Index entries for sequences related to primes in arithmetic progressions

For common differences see A088430, for initial terms see A000040, for last terms see A113834, for the APs see A231406.

For other types of APs of primes see A005115 and its comments.

a(n)=my(p=prime(n),P=prod(i=1,n-1,prime(i)),d); forprime(q=p+1,, d=q-p; if(d%P,next); for(i=2,p-1,if(!isprime(p+i*d), next(2))); return(q)) \\ Charles R Greathouse IV, Nov 08 2013

a(n) = prime(n) + A088430(n) = prime(n) + A002110(n)*A231018(n).

a(8) found by Wojciech Izykowski, from Jens Kruse Andersen, Jun 30 2014

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.

A113833 Triangle read by rows: row n (n>=2) gives a set of n primes such that the averages of all subsets are distinct primes, having the smallest largest element.

Original entry on oeis.org

3, 7, 7, 19, 67, 5, 17, 89, 1277, 209173, 322573, 536773, 1217893, 2484733

Offset: 2

N. J. A. Sloane, Jan 25 2006

If there is more than one set with the same smallest last element, choose the lexicographically earliest solution.

Note that, in each row, the n primes are equal modulo 4, 12, 12 and 120, respectively. - Row 5 from T. D. Noe, Aug 08 2006

			Triangle begins:
3, 7
7, 19, 67
5, 17, 89, 1277

Antal Balog, The prime k-tuplets conjecture on average, in "Analytic Number Theory" (eds. B. C. Berndt et al.) Birkhäuser, Boston, 1990, pp. 165-204. [Background]

Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
Andrew Granville, Prime number patterns

Cf. A113827-A113831, A113832, A113834, A088430.

Row 5 from T. D. Noe, Aug 08 2006

A165500 Maximum length of arithmetic progression starting at n such that each term k has tau(k) = tau(n).

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 2, 5, 11, 3, 13, 7, 6, 2, 17, 3, 19, 5, 7, 11

Offset: 1

Hugo van der Sanden, Sep 21 2009, Oct 09 2009

Implicitly, we require the difference d of the arithmetic progression to be positive.

a(n) <= n for all n.

			For n=4, tau(n)=3 so each term of the arithmetic progression must be the square of a prime. The difference d must be odd for n+d to qualify, in which case n+2d is even and does not qualify; so a(4)=2 is an upper bound.

Cf. A165498, A165501, A088430.

Extended to n=22 (taking advantage of A088430 for n=19) by Hugo van der Sanden, Jun 02 2015

A165501 Minimal common difference of maximal arithmetic progression starting at n such that each term k has tau(k)=tau(n).

Original entry on oeis.org

1, 2, 5, 6, 2, 150, 19, 16, 12, 1536160080, 8, 9918821194590, 300, 188, 65, 341976204789992332560, 157, 2166703103992332274919550, 24, 450, 3072320160

Offset: 2

Hugo van der Sanden, Sep 21 2009, Oct 09 2009

a(1) is not well defined, since the maximal progression has only one term.

			For n=6, A165500(n)=3, and the least difference d such that tau(6) = tau(6+d) = tau(6+2d) is d=2, so a(6)=2.

Cf. A165499, A165500, A088430.

Extended to n=22 taking advantage of A088430 for n=19, Hugo van der Sanden, Jun 02 2015

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

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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A113832 Triangle read by rows: row n (n>=2) gives a set of n primes with the property that the pairwise averages are all primes, having the smallest largest element.

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A113834 Last term of smallest p-term arithmetic progression of primes beginning with p, where p = n-th prime.

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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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A231017 Least prime q > p = prime(n) such that if d = q-p, then p, p+d, p+2d, ..., p+(p-1)d are all primes.

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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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A113833 Triangle read by rows: row n (n>=2) gives a set of n primes such that the averages of all subsets are distinct primes, having the smallest largest element.

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A165500 Maximum length of arithmetic progression starting at n such that each term k has tau(k) = tau(n).

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