A051309 -id:A051309 - cp's OEIS frontend

A000945 Euclid-Mullin sequence: a(1) = 2, a(n+1) is smallest prime factor of 1 + Product_{k=1..n} a(k).

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

Offset: 1

N. J. A. Sloane

"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 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 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. - T. D. Noe, Oct 15 2010
On Sep 11 2012, Ryan Propper factored the 256-digit number by finding a 75-digit factor by using ECM. Finding a(52) will require the factorization of a 335-digit number. See the b-file for the terms a(48) to a(51). - V. Raman, Sep 17 2012
Needs longer b-file. - N. J. A. Sloane, Dec 18 2015
A056756 gives the position of the k-th 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 (1933-2017). - Amiram Eldar, Jun 11 2021
In Ribenboim 2004, a wrong value of a(8) is given, 6221271 instead of 6221671. - Stefano Spezia, Mar 27 2025

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

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. 49-63, 1975.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 5.
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. 23-32.

Ryan Propper, Table of n, a(n) for n = 1..51 (first 47 terms from T. D. Noe)
Andrew R. Booker, On Mullin's second sequence of primes, Integers, Vol. 12, No. 6 (2012), pp. 1167-1177; arXiv preprint, arXiv:1107.3318 [math.NT], 2011-2013.
Andrew R. Booker, A variant of the Euclid-Mullin sequence containing every prime, arXiv preprint arXiv:1605.08929 [math.NT], 2016.
Andrew R. Booker and Sean A. Irvine, The Euclid-Mullin graph, Journal of Number Theory, Vol. 165 (2016), pp. 30-57; arXiv preprint, arXiv:1508.03039 [math.NT], 2015-2016.
Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Vol. 56(104), No. 1 (2013), pp. 73-98.
Keith Conrad, The infinitude of the primes, University of Connecticut, 2020.
C. D. Cox and A. J. van der Poorten, On a sequence of prime numbers, Journal of the Australian Mathematical Society, Vol. 8 (1968), pp. 571-574.
FactorDB, Status of EM51.
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).
Robert R. Korfhage, On a sequence of prime numbers, Bull Amer. Math. Soc., Vol. 70 (1964), pp. 341, 342, 747. [Annotated scanned copy]
Evelyn Lamb, A Curious Sequence of Prime Numbers, Scientific American blog (2019).
Des MacHale, Infinitely many proofs that there are infinitely many primes, Math. Gazette, Vol. 97, No. 540 (2013), pp. 495-498.
Mersenne Forum, Factoring 43rd Term of Euclid-Mullin sequence.
Mersenne Forum, Factoring EM47.
Romeo Meštrović, 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.
Albert A. Mullin, Research Problem 8: Recursive function theory, Bull. Amer. Math. Soc., Vol. 69, No. 6 (1963), p. 737.
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]
OEIS wiki, OEIS sequences needing factors
Paul Pollack and Enrique Treviño, The Primes that Euclid Forgot, Amer. Math. Monthly, Vol. 121, No. 5 (2014), pp. 433-437. MR3193727; alternative link.
Daphne Stouthart, Euclid and the infinite number of missing primes, Bachelor Thesis, Utrecht Univ (Netherlands, 2024). See p. 1.
Samuel S. Wagstaff, Jr., Emails to N. J. A. Sloane, May 30 1991.
Samuel S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, Vol. 8 (1993), pp. 23-32. (Annotated scanned copy)

Cf. A000946, A005265, A005266, A051309-A051334, A051614, A051615, A051616, A056756.

a :=n-> if n = 1 then 2 else numtheory:-divisors(mul(a(i),i = 1 .. n-1)+1)[2] fi: seq(a(n), n=1..15);
# Robert FERREOL, Sep 25 2019

f[1]=2; f[n_] := f[n] = FactorInteger[Product[f[i], {i, 1, n - 1}] + 1][[1, 1]]; Table[f[n], {n, 1, 46}]
nxt[{p_,a_}]:=With[{c=FactorInteger[p+1][[1,1]]},{p*c,c}]; Rest[NestList[nxt,{1,2},20][[;;,2]]] (* Harvey P. Dale, Feb 02 2025 *)

print1(k=2);for(n=2,20,print1(", ",p=factor(k+1)[1,1]);k*=p) \\ Charles R Greathouse IV, Jun 10 2011

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

A093777 a(n) is the smallest prime which, if used to start a Euclid-Mullin sequence (like A000945), the resulting sequence contains the n consecutive primes 2, 3, ..., prime(n).

Original entry on oeis.org

2, 2, 19, 199, 2089, 99109, 1960969, 10129129, 87726649, 4549584049, 328034245549, 20584643748679, 666188861477149, 31395465477725359, 894857713367947339, 434392154438254391389, 17934770256689308411399

Offset: 1

Labos Elemer, May 03 2004

nonn

Thanks in part to Dirichlet's theorem, a(n) exists for each n. - Don Reble, Oct 07 2006

			a(1) = a(2) = 2 because they generate {2,3,7,43,13,...};
a(3) = 19 because it generates {19,2,3,5,571,271,...}, see A051312;
a(4) = 199 because it generates {199,2,3,5,7,23,881,...};
a(5) = 2089 because it generates {2089,2,3,5,7,11,269,...};
a(6) = 99109 because it generates {99109,2,3,5,7,11,13,2976243271,...};
a(7) = 1960969 because it generates {1960969,2,3,5,7,11,13,17,281,47,419,5539788476533581271,37,19,173,...}

Toshitaka Suzuki, Table of n, a(n) for n = 1..100

Cf. A093778, A093779, A093780, A000945, A051309-A051334, A051614-A051616, A056756.

More terms from Don Reble, Oct 07 2006

cp's OEIS Frontend

A000945 Euclid-Mullin sequence: a(1) = 2, a(n+1) is smallest prime factor of 1 + Product_{k=1..n} a(k).

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A093777 a(n) is the smallest prime which, if used to start a Euclid-Mullin sequence (like A000945), the resulting sequence contains the n consecutive primes 2, 3, ..., prime(n).

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A051616 Primes p for which A051614(p) is 3 but are not Sophie Germain primes.

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A094153 a(n) is least prime p such that 7 is the n-th term in the Euclid-Mullin sequence starting at p, or 0 if no such prime p exists.

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A051615 Primes for which A051614 is 3.

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