A005266 a(1)=3, b(n) = Product_{k=1..n} a(k), a(n+1) is the largest prime factor of (b(n)-1). (Formerly M2247)

3, 2, 5, 29, 79, 68729, 3739, 6221191, 157170297801581, 70724343608203457341903, 46316297682014731387158877659877, 78592684042614093322289223662773, 181891012640244955605725966274974474087, 547275580337664165337990140111772164867508038795347198579326533639132704344301831464707648235639448747816483406685904347568344407941

Offset

1,1

Comments

Suggested by Euclid's proof that there are infinitely many primes.

a(15) requires completing the factorization: 13 * 67 * 14479 * 167197 * 924769 * 2688244927 * 888838110930755119 * 14372541055015356634061816579965403 * C211 where C211=6609133306626483634448666494646737799624640616060730302142187545405582531010390290502001156883917023202671554510633460047901459959959325342475132778791495112937562941066523907603281586796876335607258627832127303. - Sean A. Irvine, Nov 10 2009

References

R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.

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

S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.

Links

Table of n, a(n) for n=1..14.

R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.

Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.

S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)

Status of a(15) in factordb.com.

Maple

with(numtheory):
a:= proc(n) option remember;
      `if`(n=1, 3, max(factorset(mul(a(i), i=1..n-1)-1)[]))
    end:
seq(a(n), n=1..10);  # Alois P. Heinz, Sep 26 2013

Mathematica

a[0] = 3; a[n_] := a[n] = Block[{p = Times @@ (a[#] & /@ Range[0, n - 1]) - 1}, FactorInteger[p][[-1, 1]]]; Array[a, 13] (* Robert G. Wilson v, Sep 26 2013 *)

Crossrefs

Cf. A000945, A000946, A005265.

Essentially the same as A084599.

Sequence in context: A302854 A248243 A005265 * A005267 A209269 A373562

Adjacent sequences: A005263 A005264 A005265 * A005267 A005268 A005269

Keyword

nonn,nice,hard

Author

N. J. A. Sloane

Extensions

a(14) from Joe K. Crump (joecr(AT)carolina.rr.com), Jul 26 2000

Status

approved