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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.

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).


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.

A. R. Booker, S. A. Irvine, The Euclid-Mullin graph, arXiv preprint arXiv:1508.03039 [math.NT], 2015.

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.

R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.

R. R. Khorfhage, On a sequence of prime numbers, Bull Amer. Math. Soc., 70 (1964), pp. 341, 342, 747. [Annotated scanned copy]

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

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 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012

A. A. Mullin, Research Problem 8: Recursive function theory, Bull. Amer. Math. Soc., 69 (1963), 737.

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

Thorkil Naur, Letter to N. J. A. Sloane, Aug 27 1991, together with copies of "Mullin's sequence of primes is not monotonic" (1984) and "New integer factorizations" (1983) [Annotated scanned copies]

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

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

S. S. Wagstaff, Jr., Emails to N. J. A. Sloane, May 30 1991

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

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


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 *)


(PARI) gpf(n)=my(f=factor(n)[, 1]); f[#f];

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


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 12 19:33 EST 2018. Contains 318081 sequences. (Running on oeis4.)