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

%I A126263 #60 Sep 09 2024 16:07:31
%S A126263 2,3,7,43,13,139,3263443,547,607,1033,31051,29881,67003,9119521,
%T A126263 6212157481,5295435634831,31401519357481261,77366930214021991992277,
%U A126263 181,1987,112374829138729,114152531605972711,35874380272246624152764569191134894955972560447869169859142453622851
%N A126263 List of primes generated by factoring successive integers in Sylvester's sequence (A000058).
%C A126263 The list is infinite and no term repeats since Sylvester's sequence is an infinite coprime sequence.
%C A126263 However, it appears to be unknown whether all terms in A000058 are squarefree. - _Jeppe Stig Nielsen_, Apr 23 2020
%D A126263 Barry Mazur and William Stein, Prime Numbers and the Riemann Hypothesis, Cambridge University Press, 2016. See p. 9.
%H A126263 Ray Chandler, <a href="/A126263/b126263.txt">Table of n, a(n) for n = 1..28</a> (first 27 terms from William Stein)
%H A126263 J. K. Andersen, <a href="http://primerecords.dk/sylvester-factors.htm">Factorization of Sylvester's sequence</a>.
%H A126263 Filip Saidak, <a href="https://t5k.org/notes/proofs/infinite/Saidak.html">Proof of Euclid's Theorem</a>.
%H A126263 Filip Saidak, <a href="http://www.jstor.org/stable/27642094">A New Proof of Euclid's Theorem</a>, Amer. Math. Monthly, Dec. 2006.
%e A126263 2 = 2, 3 = 3, 7 = 7, 43 = 43, 1807 = 13 * 139, 3263443 = 3263443,
%e A126263 10650056950807 = 547 * 607 * 1033 * 31051,
%e A126263 113423713055421844361000443 = 29881 * 67003 * 9119521 * 6212157481,
%e A126263 12864938683278671740537145998360961546653259485195807 = 5295435634831 * 31401519357481261 * 77366930214021991992277.
%e A126263 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443 = 181 * 1987 * 112374829138729 * 114152531605972711 * 35874380272246624152764569191134894955972560447869169859142453622851. - _Jonathan Sondow_, Jan 26 2014
%p A126263 a(0):=2; for n from 0 to 8 do a(n+1):=a(n)^2-a(n)+1;ifactor(%); od;
%t A126263 Flatten[FactorInteger[NestList[#^2 - # + 1 &, 2, 8]][[All, All, 1]]] (* _Paolo Xausa_, Sep 09 2024 *)
%o A126263 (Sage)
%o A126263 v = [2]
%o A126263 for n in range(12):
%o A126263     v.append(v[-1]^2-v[-1]+1)
%o A126263     print(prime_divisors(v[-1])) # William Stein, Aug 26 2009
%o A126263 (PARI)
%o A126263 v=[2]; for(i=1,10, v=concat(v,Set(factor(vecprod(v)+1)[,1]))); v \\ _Charles R Greathouse IV_, Oct 02 2014
%Y A126263 Cf. A000058, A007996, A236433.
%K A126263 nonn
%O A126263 1,1
%A A126263 Howard L. Warth (hlw6c2(AT)umr.edu), Dec 22 2006
%E A126263 Offset corrected by _N. J. A. Sloane_, Aug 20 2009
%E A126263 a(23)-a(27) from William Stein (wstein(AT)gmail.com), Aug 20 2009, Aug 21 2009
%E A126263 a(17) corrected by _D. S. McNeil_, Dec 10 2010
%E A126263 b-file updated at the suggestion of _Hans Havermann_ by _Ray Chandler_, Feb 27 2015