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Search: id:A000945
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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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+0
78
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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; listen)
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OFFSET
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1,1
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COMMENT
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"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
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REFERENCES
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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.
S. S. Wagstaff, Jr., Computing Euclid's primes, Bull. Institute Combin. Applications, 8 (1993), 23-32.
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EXAMPLE
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a(5) is equal to 13 because 2*3*7*43+1 = 1807 = 13 * 139.
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MATHEMATICA
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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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CROSSREFS
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Cf. A000946, A005265, A005266.
Cf. A051309-A051334, A051614, A051614-A051616, A056756.
Adjacent sequences: A000942 A000943 A000944 this_sequence A000946 A000947 A000948
Sequence in context: A037843 A102604 A119662 this_sequence A126263 A030087 A106864
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KEYWORD
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nonn,nice,hard
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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