A126263 List of primes generated by factoring successive integers in Sylvester's sequence (A000058).

2, 3, 7, 43, 13, 139, 3263443, 547, 607, 1033, 31051, 29881, 67003, 9119521, 6212157481, 5295435634831, 31401519357481261, 77366930214021991992277, 181, 1987, 112374829138729, 114152531605972711, 35874380272246624152764569191134894955972560447869169859142453622851

Offset

1,1

Comments

The list is infinite and no term repeats since Sylvester's sequence is an infinite coprime sequence.

However, it appears to be unknown whether all terms in A000058 are squarefree. - Jeppe Stig Nielsen, Apr 23 2020

References

Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, 2016. See p. 9.

Links

Ray Chandler, Table of n, a(n) for n = 1..28 (first 27 terms from William Stein)

J. K. Andersen, Factorization of Sylvester's sequence.

Filip Saidak, Proof of Euclid's Theorem.

Filip Saidak, A New Proof of Euclid's Theorem, Amer. Math. Monthly, Dec. 2006.

Example

2 = 2, 3 = 3, 7 = 7, 43 = 43, 1807 = 13 * 139, 3263443 = 3263443,
10650056950807 = 547 * 607 * 1033 * 31051,
113423713055421844361000443 = 29881 * 67003 * 9119521 * 6212157481,
12864938683278671740537145998360961546653259485195807 = 5295435634831 * 31401519357481261 * 77366930214021991992277.
165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 = 181 * 1987 * 112374829138729 * 114152531605972711 * 35874380272246624152764569191134894955972560447869169859142453622851. - Jonathan Sondow, Jan 26 2014

Maple

a(0):=2; for n from 0 to 8 do a(n+1):=a(n)^2-a(n)+1; ifactor(%); od;

Prog

(Sage)
v = [2]
for n in range(12):
    v.append(v[-1]^2-v[-1]+1)
    print(prime_divisors(v[-1])) # William Stein, Aug 26 2009
(PARI)
v=[2]; for(i=1, 10, v=concat(v, Set(factor(vecprod(v)+1)[, 1]))); v \\ Charles R Greathouse IV, Oct 02 2014

Crossrefs

Cf. A000058, A007996, A236433.

Sequence in context: A260819 A000945 A261564 * A323605 A354376 A216826

Adjacent sequences: A126260 A126261 A126262 * A126264 A126265 A126266

Keyword

nonn

Author

Howard L. Warth (hlw6c2(AT)umr.edu), Dec 22 2006

Extensions

Offset corrected by N. J. A. Sloane, Aug 20 2009

a(23)-a(27) from William Stein (wstein(AT)gmail.com), Aug 20 2009, Aug 21 2009

a(17) corrected by D. S. McNeil, Dec 10 2010

b-file updated at the suggestion of Hans Havermann by Ray Chandler, Feb 27 2015

Status

approved