

A000946


EuclidMullin sequence: a(1) = 2, a(n+1) is largest prime factor of Product_{k=1..n} a(k) + 1.
(Formerly M0864 N0330)


50



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

1,1


COMMENTS

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."


REFERENCES

R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 4963, 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).


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.
C. Cobeli and A. Zaharescu, Promenade around Pascal TriangleNumber Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, pp. 7398.
C. D. Cox and A. J. van der Poorten, On a sequence of prime numbers, Journal of the Australian Mathematical Society 8 (1968), pp. 571574.
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), 495498.
Mersenne Forum, The second EuclidMullin sequence
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, arXiv preprint arXiv:1202.3670, 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), 4344.
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, 433437. 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), 2332.


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 *)


PROG

(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, i1, v[j]))); v; \\ Anders Hellström, Aug 14 2015


CROSSREFS

Cf. A000945, A005265, A005266.
Sequence in context: A218467 A241166 A083369 * A091771 A072714 A051786
Adjacent sequences: A000943 A000944 A000945 * A000947 A000948 A000949


KEYWORD

nonn,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

Extended by Andrew R. Booker, Mar 13 2013


STATUS

approved



