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A126263 List of primes generated by factoring successive integers in Sylvester's sequence (A000058). 6
2, 3, 7, 43, 13, 139, 3263443, 547, 607, 1033, 31051, 29881, 67003, 9119521, 6212157481, 5295435634831, 31401519357481261, 77366930214021991992277, 181, 1987, 112374829138729, 114152531605972711, 35874380272246624152764569191134894955972560447869169859142453622851 (list; graph; refs; listen; history; text; internal format)
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 if all terms in A000058 are squarefree. - Jeppe Stig Nielsen, Apr 23 2020

LINKS

William Stein and 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 A216826 A030087

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 suggestion of Hans Havermann by Ray Chandler, Feb 27 2015

STATUS

approved

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Last modified April 21 15:48 EDT 2021. Contains 343154 sequences. (Running on oeis4.)