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

Author

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

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

Cf. A000058, A007996, A236433.

Programs

Maple

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

Mathematica

Flatten[FactorInteger[NestList[#^2 - # + 1 &, 2, 8]][[All, All, 1]]] (* Paolo Xausa, Sep 09 2024 *)

PARI

v=[2]; for(i=1,10, v=concat(v,Set(factor(vecprod(v)+1)[,1]))); v \\ Charles R Greathouse IV, Oct 02 2014

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

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