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A000946 Euclid-Mullin sequence: a(1) = 2, a(n+1) is largest prime factor of Product_{k=1..n} a(k) + 1.
(Formerly M0864 N0330)
2, 3, 7, 43, 139, 50207, 340999, 2365347734339, 4680225641471129, 1368845206580129, 889340324577880670089824574922371, 20766142440959799312827873190033784610984957267051218394040721 (list; graph; refs; listen; history; text; internal format)



Cox and van der Poorten show that 5, 11, 13, 17, ... are not members of this sequence. - Charles R Greathouse IV, Jul 02 2007

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."


R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.

Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495-498.

T. Naur, Mullin's sequence of primes is not monotonic, Proc. Amer. Math. Soc., 90 (1984), 43-44.

Pollack, Paul; TreviƱo, Enrique. The Primes that Euclid Forgot. Amer. Math. Monthly 121 (2014), no. 5, 433--437. MR3193727

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.


T. D. Noe, Table of n, a(n) for n = 1..14

Andrew R. Booker, On Mullin's second sequence of primes, Integers, 12A (2012), article A4.

C. Cobeli and A. Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, pp. 73-98.

C. D. Cox and A. J. van der Poorten, On a sequence of prime numbers, Journal of the Australian Mathematical Society 8 (1968), pp. 571-574.

Mersenne Forum, The second Euclid-Mullin sequence

R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012

P. Pollack and E. Trevino, The primes that Euclid forgot, 2013. - From N. J. A. Sloane, Feb 20 2013


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 *)


Cf. A000945, A005265, A005266.

Sequence in context: A218467 A241166 A083369 * A091771 A072714 A051786

Adjacent sequences:  A000943 A000944 A000945 * A000947 A000948 A000949




N. J. A. Sloane.


Extended by Andrew R. Booker, Mar 13 2013



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Last modified December 22 16:00 EST 2014. Contains 252364 sequences.