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%I M0864 N0330 #112 May 08 2023 02:08:57
%S 2,3,7,43,139,50207,340999,2365347734339,4680225641471129,
%T 1368845206580129,889340324577880670089824574922371,
%U 20766142440959799312827873190033784610984957267051218394040721
%N Euclid-Mullin sequence: a(1) = 2, a(n+1) is the largest prime factor of 1 + Product_{k=1..n} a(k).
%C Cox and van der Poorten show that 5, 11, 13, 17, ... are not members of this sequence. - _Charles R Greathouse IV_, Jul 02 2007
%C Booker's abstract claims: "We consider the second of Mullin's sequences of prime numbers related to Euclid's proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten."
%D R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A000946/b000946.txt">Table of n, a(n) for n = 1..14</a>
%H Andrew R. Booker, <a href="http://www.emis.de/journals/INTEGERS/papers/a4self/a4self.Abstract.html">On Mullin's second sequence of primes</a>, Integers, 12A (2012), article A4.
%H A. R. Booker, S. A. Irvine, <a href="http://arxiv.org/abs/1508.03039">The Euclid-Mullin graph</a>, arXiv preprint arXiv:1508.03039 [math.NT], 2015.
%H C. Cobeli and A. Zaharescu, <a href="http://rms.unibuc.ro/bulletin/pdf/56-1/PromenadePascalPart1.pdf">Promenade around Pascal Triangle-Number Motives</a>, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, pp. 73-98.
%H C. D. Cox and A. J. van der Poorten, <a href="http://dx.doi.org/10.1017/S1446788700006236">On a sequence of prime numbers</a>, Journal of the Australian Mathematical Society 8 (1968), pp. 571-574.
%H R. K. Guy and R. Nowakowski, <a href="/A000945/a000945_5.pdf">Discovering primes with Euclid</a>, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
%H R. R. Khorfhage, <a href="/A000945/a000945_3.pdf">On a sequence of prime numbers</a>, Bull Amer. Math. Soc., 70 (1964), pp. 341, 342, 747. [Annotated scanned copy]
%H Des MacHale, <a href="http://dx.doi.org/10.2307/3621650">Infinitely many proofs that there are infinitely many primes</a>, Math. Gazette, 97 (No. 540, 2013), 495-498.
%H Mersenne Forum, <a href="http://mersenneforum.org/showthread.php?t=17884">The second Euclid-Mullin sequence</a>
%H R. Mestrovic, <a href="http://arxiv.org/abs/1202.3670">Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof</a>, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012
%H A. A. Mullin, <a href="http://dx.doi.org/10.1090/S0002-9904-1963-11017-4">Research Problem 8: Recursive function theory</a>, Bull. Amer. Math. Soc., 69 (1963), 737.
%H Thorkil Naur, <a href="http://dx.doi.org/10.1090/S0002-9939-1984-0722412-X">Mullin's sequence of primes is not monotonic</a>, Proc. Amer. Math. Soc., 90 (1984), 43-44.
%H Thorkil Naur, <a href="/A000945/a000945.pdf">Letter to N. J. A. Sloane, Aug 27 1991</a>, together with copies of "Mullin's sequence of primes is not monotonic" (1984) and "New integer factorizations" (1983) [Annotated scanned copies]
%H Paul Pollack and Enrique Treviño, <a href="http://web.archive.org/web/20171110181550/http://alpha.math.uga.edu/~pollack/mullin.pdf">The primes that Euclid forgot</a>, 2013. - From _N. J. A. Sloane_, Feb 20 2013
%H Paul Pollack and Enrique Treviño, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.05.433">The Primes that Euclid Forgot</a>, Amer. Math. Monthly 121 (2014), no. 5, 433--437. MR3193727
%H S. S. Wagstaff, Jr., <a href="/A000945/a000945_1.pdf">Emails to N. J. A. Sloane, May 30 1991</a>
%H S. S. Wagstaff, Jr., <a href="http://web.archive.org/web/20120227155112/http://homes.cerias.purdue.edu/~ssw/euc.pdf">Computing Euclid's primes</a>, Bull. Institute Combin. Applications, 8 (1993), 23-32.
%H S. S. Wagstaff, Jr., <a href="/A000945/a000945_4.pdf">Computing Euclid's primes</a>, Bull. Institute Combin. Applications, 8 (1993), 23-32. (Annotated scanned copy)
%t f[1] = 2; f[n_] := f[n] = FactorInteger[Product[f[i], {i, 1, n - 1}] + 1][[-1, 1]]; Table[f[n], {n, 1, 10}] (* _Alonso del Arte_, Jun 25 2011 based on the program given for A000945 *)
%o (PARI) gpf(n)=my(f=factor(n)[, 1]);f[#f];
%o first(m)=my(v=vector(m));v[1]=2;for(i=2,m,v[i]=gpf(1+prod(j=1,i-1,v[j])));v; \\ _Anders Hellström_, Aug 14 2015
%Y Cf. A000945, A005265, A005266, A216227.
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_
%E Extended by _Andrew R. Booker_, Mar 13 2013
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