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

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Comments

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.

Examples

			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.
		

References

• 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).

Links

• Jens Kruse Andersen, Primes in Arithmetic Progression Records [May have candidates for later terms in this sequence.]
• H. Dubner and H. 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.
• A. Moran, P. Pritchard and A. Thyssen, Twenty-two primes in arithmetic progression, Math. Comp.64 (1995), no.211, 1337-1339.
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

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

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

Programs

Mathematica

(* 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}]

Formula

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

Extensions

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