VOOZH about

URL: https://oeis.org/A126263

⇱ A126263 - OEIS

login
A126263
List of primes generated by factoring successive integers in Sylvester's sequence (A000058).
7
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)
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;
MATHEMATICA
Flatten[FactorInteger[NestList[#^2 - # + 1 &, 2, 8]][[All, All, 1]]] (* Paolo Xausa, Sep 09 2024 *)
PROG
(SageMath)
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
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