A053647 First term of first sequence of n primes in arithmetic progression with a common difference equal to the product of first n primes.

2, 5, 7, 13, 37, 73, 7937, 7703, 272809, 640943, 5378959, 116137159, 3708797237, 114649314209, 158317270283

Offset

1,1

Comments

a(14) > 2^32 and a(15) > 2^32. - Jud McCranie

Links

Table of n, a(n) for n=1..15.

R. Chapman, Dirichlet's theorem: a real variable approach, 2008.

B. Green & T. Tao, The primes contain arbitrarily long arithmetic progressions, arXiv:math/0404188 [math.NT], 2004-2007.

Index entries for sequences related to primes in arithmetic progressions

Example

For n=3, product of first 3 primes is 30. The first arithmetic progression of 3 primes with difference 30 starts at 7 (7, 37, 67), so a(3)=7.

Mathematica

(* This program is not convenient beyond 10 terms *) r[p1_, n_] := Reduce[p[1] = p1; Equal @@ Append[Table[p[k + 1] - p[k], {k, 1, n - 1}], Product[Prime[k], {k, 1, n}]], p[2], Primes]; a[n_] := a[n] = Catch[For[k = 1, k <= 10^5, k++, If[r[p1 = Prime[k], n] =!= False, Throw[p1]]]]; Table[Print[a[n]]; a[n], {n, 1, 10}] (* Jean-François Alcover, Dec 27 2012 *)

Crossrefs

Sequence in context: A177997 A238776 A141112 * A023242 A164570 A348671

Adjacent sequences: A053644 A053645 A053646 * A053648 A053649 A053650

Keyword

hard,nonn,nice

Author

G. L. Honaker, Jr., Feb 18 2000

Extensions

Last 3 terms from Jud McCranie, Feb 28 2000

a(14)-a(15) from Donovan Johnson, Oct 20 2009

Status

approved