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A000945
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Euclid-Mullin sequence: a(1) = 2, a(n+1) is smallest prime factor of Product_{k=1..n} a(k) + 1.
(Formerly M0863 N0329)
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81
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2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, 2801, 11, 17, 5471, 52662739, 23003, 30693651606209, 37, 1741, 1313797957, 887, 71, 7127, 109, 23, 97, 159227, 643679794963466223081509857, 103, 1079990819, 9539, 3143065813, 29, 3847, 89, 19, 577, 223, 139703, 457, 9649, 61, 4357
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| "Does the sequence ... contain every prime? ... [It] was considered by Guy and Nowakowski and later by Shanks, [Wagstaff 1993] computed the sequence through the 43rd term. The computational problem inherent in continuing the sequence further is the enormous size of the numbers that must be factored. Already the number a(1)* ... *a(43) + 1 has 180 digits." - Crandall and Pomerance.
If this variant of Euclid-Mullin sequence is initiated either with 3, 7 or 43 instead of 2, then from a[5] onwards it is unchanged. See also A051614. - Labos E. (labos(AT)ana.sote.hu), May 03 2004
Wilfrid Keller informed me that a(1)* ... *a(43) + 1 was factored as the product of two primes on March 9, 2010 by the GNFS method. See the post in the Mersenne Forum for more details. The smaller 68-digit prime is a(44). Terms a(45)-a(47) were easy to find. Finding a(48) will require the factorization of a 256-digit number. See the b-file for the four new terms. [From T. D. Noe (noe(AT)sspectra.com), Oct 15 2010]
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REFERENCES
| A. R. Booker, On Mullin's second sequence of primes, Arxiv preprint arXiv:1107.3318, 2011
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 6.
R. K. Guy and R. Nowakowski, Discovering primes with Euclid, Delta (Waukesha), Vol. 5, pp. 49-63, 1975.
A. A. Mullin, 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.
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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..47
Mersenne Forum, Factoring 43rd Term of Euclid-Mullin sequence [From T. D. Noe (noe(AT)sspectra.com), Oct 15 2010]
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EXAMPLE
| a(5) is equal to 13 because 2*3*7*43+1 = 1807 = 13 * 139.
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MATHEMATICA
| f[1]=2; f[n_] := f[n] = FactorInteger[Product[f[i], {i, 1, n - 1}] + 1][[1, 1]] Table[f[n], {n, 1, 46}]
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PROG
| (PARI) print1(k=2); for(n=2, 20, print1(", ", p=factor(k+1)[1, 1]); k*=p) \\ Charles R Greathouse IV, Jun 10 2011
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CROSSREFS
| Cf. A000946, A005265, A005266, A051309-A051334, A051614, A051614-A051616, A056756.
Sequence in context: A102604 A119662 A163157 * A126263 A030087 A106864
Adjacent sequences: A000942 A000943 A000944 * A000946 A000947 A000948
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KEYWORD
| nonn,nice,hard
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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