

A000945


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


101



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(51).  V. Raman, Sep 17 2012
A056756 gives the position of the kth prime in this sequence for each k.  Jianing Song, May 07 2021
Named after the Greek mathematician Euclid (flourished c. 300 B.C.) and the American engineer and mathematician Albert Alkins Mullin (19332017).  Amiram Eldar, Jun 11 2021


REFERENCES

Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 6.
Richard Guy and Richard 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).
Samuel S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, Vol. 8 (1993), pp. 2332.


LINKS

Richard Guy and Richard Nowakowski, Discovering primes with Euclid, Research Paper No. 260 (Nov 1974), The University of Calgary Department of Mathematics, Statistics and Computing Science.
Lucas Hoogendijk, Prime Generators, Bachelor Thesis, Utrecht University (Netherlands, 2020).
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]
Samuel S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, Vol. 8 (1993), pp. 2332. (Annotated scanned copy)


EXAMPLE

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


MAPLE

a :=n> if n = 1 then 2 else numtheory:divisors(mul(a(i), i = 1 .. n1)+1)[2] fi: seq(a(n), n=1..15);


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) P=[]; until(, print(P=concat(P, factor(vecprod(P)+1)[1, 1]))) \\ Jeppe Stig Nielsen, Apr 01 2024


CROSSREFS



KEYWORD

nonn,nice,hard


AUTHOR



STATUS

approved



