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A000946 Euclid-Mullin sequence: a(1) = 2, a(n+1) is the largest prime factor of 1 + Product_{k=1..n} a(k).
(Formerly M0864 N0330)
53
2, 3, 7, 43, 139, 50207, 340999, 2365347734339, 4680225641471129, 1368845206580129, 889340324577880670089824574922371, 20766142440959799312827873190033784610984957267051218394040721 (list; graph; refs; listen; history; text; internal format)
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. 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).
LINKS
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.
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.
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. - From 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
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)
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, i-1, v[j]))); v; \\ Anders Hellström, Aug 14 2015
CROSSREFS
Sequence in context: A218467 A241166 A083369 * A367020 A091771 A339806
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Extended by Andrew R. Booker, Mar 13 2013
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)