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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.
(Formerly M0854)
2, 3, 7, 23, 29, 157, 907, 1669, 1879, 2089, 249037, 262897, 725663, 36850999, 173471351, 198793279, 4827507229, 17010526363, 83547839407, 572945039351, 6269243827111 (list; graph; refs; listen; history; text; internal format)



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.


H. Dubner and H. Nelson, Seven consecutive primes in arithmetic progression, Math. Comp., 66 (1997) 1743-1749. MR 98a:11122.

R. K. Guy, Unsolved Problems in Number Theory, A5.

A. Moran, P. Pritchard and A. Thyssen, Twenty-two primes in arithmetic progression, Math. Comp.64 (1995), no.211, 1337-1339.

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


Table of n, a(n) for n=1..21.

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

Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics 167 (2008), pp. 481-547. arXiv:math/0404188

Ben Green and Terence Tao, A bound for progressions of length k in the primes

Andrew Granville, Prime number patterns

Index entries for sequences related to primes in arithmetic progressions


Green and 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


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.


(* 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}]


For the associated gaps see A093364, for the initial terms see A113827. Cf. A006560, A096003.

Cf. A113830-A113834, A088430.

Sequence in context: A156615 A158054 A134412 * A113872 A120302 A093363

Adjacent sequences:  A005112 A005113 A005114 * A005116 A005117 A005118




N. J. A. Sloane.


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 (djr(AT)nk.ca), 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



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Last modified November 29 10:31 EST 2015. Contains 264643 sequences.