

A000945


EuclidMullin sequence: a(1) = 2, a(n+1) is smallest prime factor of Product_{k=1..n} a(k) + 1.
(Formerly M0863 N0329)


87



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
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OFFSET

1,1


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 EuclidMullin sequence is initiated either with 3, 7 or 43 instead of 2, then from a[5] onwards it is unchanged. See also A051614.  Labos Elemer, May 03 2004
Wilfrid Keller informed me that a(1)* ... *a(43) + 1 was factored as the product of two primes on Mar 09 2010 by the GNFS method. See the post in the Mersenne Forum for more details. The smaller 68digit prime is a(44). Terms a(45)a(47) were easy to find. Finding a(48) will require the factorization of a 256digit number. See the bfile for the four new terms.  T. D. Noe, Oct 15 2010
On Sep 11 2012, Ryan Propper factored the 256digit number by finding a 75digit factor by using ECM. Finding a(52) will require the factorization of a 335digit number. See the bfile for the terms a(48) to a(52).  V. Raman, Sep 17 2012


REFERENCES

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. 4963, 1975.
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, 97 (No. 540, 2013), 495498.
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), 4344.
Pollack, Paul; TreviĆ±o, Enrique. The Primes that Euclid Forgot. Amer. Math. Monthly 121 (2014), no. 5, 433437. MR3193727
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), 2332.


LINKS

T. D. Noe and Ryan Propper, Table of n, a(n) for n = 1..51 (first 47 terms from T. D. Noe)
C. Cobeli and A. Zaharescu, Promenade around Pascal TriangleNumber Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, 7398.
P. Pollack and E. Trevino, The primes that Euclid forgot, 2013.
A. R. Booker, On Mullin's second sequence of primes, arXiv preprint arXiv:1107.3318, 2011
Mersenne Forum, Factoring 43rd Term of EuclidMullin sequence
Mersenne Forum, Factoring EM47
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
OEIS wiki, OEIS sequences needing factors


EXAMPLE

a(5) is equal to 13 because 2*3*7*43+1 = 1807 = 13 * 139.


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}]


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


CROSSREFS

Cf. A000946, A005265, A005266, A051309A051334, A051614, A051615, A051616, A056756.
Sequence in context: A102604 A119662 A163157 * A126263 A216826 A030087
Adjacent sequences: A000942 A000943 A000944 * A000946 A000947 A000948


KEYWORD

nonn,nice,hard,changed


AUTHOR

N. J. A. Sloane


STATUS

approved



