A000946 Euclid-Mullin sequence: a(1) = 2, a(n+1) is the largest prime factor of 1 + Product_{k=1..n} a(k).

2, 3, 7, 43, 139, 50207, 340999, 2365347734339, 4680225641471129, 1368845206580129, 889340324577880670089824574922371, 20766142440959799312827873190033784610984957267051218394040721

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Cox and van der Poorten show that 5, 11, 13, 17, ... (A216227) 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."

References

• R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 5.
• 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).

Links

• 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.
• Andrew R. Booker and Sean 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
• Romeo Meštrović, 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. - _N. J. A. Sloane_, Jun 13 2012
• A. A. Mullin, Research Problem 8: Recursive function theory, Bull. Amer. Math. Soc., 69 (1963), 737.
• Thorkil 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]
• Paul Pollack and Enrique Treviño, The primes that Euclid forgot, 2013. - _N. J. A. Sloane_, Feb 20 2013
• Paul Pollack and Enrique Treviño, The Primes that Euclid Forgot, Amer. Math. Monthly 121 (2014), no. 5, 433-437. MR3193727
• Daphne Stouthart, Euclid and the infinite number of missing primes, Bachelor Thesis, Utrecht Univ (Netherlands, 2024). See p. 1.
• 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)

Crossrefs

Cf. A000945, A005265, A005266, A216227.

Programs

Mathematica

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

Extensions

Extended by Andrew R. Booker, Mar 13 2013