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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard

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

• 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.

Crossrefs

Cf. A000945, A000946, A005265.

Essentially the same as A084599.

Programs

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 *)

Extensions

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