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A002072
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a(n) = smallest number m such that for all k > m, either k or k+1 has a prime factor > prime(n).
(Formerly M4560 N1942)
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22
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1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 11859210, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125, 1453579866024, 20628591204480, 31887350832896, 31887350832896, 119089041053696, 2286831727304144, 9591468737351909375, 9591468737351909375, 9591468737351909375, 9591468737351909375, 9591468737351909375, 19316158377073923834000
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OFFSET
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1,2
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COMMENTS
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An effective abc conjecture (c < rad(abc)^2) would imply that a(27) = a(28) = ... = a(32), and a(33) = 124225935845233319439173. - Lucas A. Brown, Sep 20 2020
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REFERENCES
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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).
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LINKS
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Table of n, a(n) for n=1..27.
E. F. Ecklund and R. B. Eggleton, Prime factors of consecutive integers, Amer. Math. Monthly, 79 (1972), 1082-1089.
D. H. Lehmer, On a problem of Størmer, Ill. J. Math., 8 (1964), 57-79.
Don Reble, Python program
Jim White, Results to P = 127
Wikipedia, Størmer's theorem
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FORMULA
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a(n) < 10^n/n except for n=4. (Conjectured, from experimental data.) - M. F. Hasler, Jan 16 2015
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EXAMPLE
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a(1) = 1 since for any number k greater than 1, it is impossible that k and k+1 both are powers of 2, so at least one of them has a prime factor > 2. (For m = 0 this would not hold for k = 1, k+1 = 2.)
a(2) = 8 since for any larger k, we cannot have k and k+1 both 3-smooth (cf. A003586).
31887350832897 = 3^9*7*37*41^2*61^2, 31887350832896 = 2^8*13*19*23*29^4*31, this number appears twice because there is no pair of numbers with max. factor = 67 that is larger than this number.
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MATHEMATICA
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smoothNumbers[p_?PrimeQ, max_Integer] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand[Log[pp[[j]], max/Times @@ (Take[pp, j-1]^Take[aa, j-1])]] }, {j, 1, k}]; Sort[Flatten[Table[Times @@ (pp^aa), Evaluate[ Sequence @@ iter]]]]]; a[n_] := Module[{sn = smoothNumbers[Prime[n], Ceiling[2000 + 10^n/n]], pos}, pos = Position[Differences[sn], 1][[-1, 1]]; sn[[pos]]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Nov 17 2016, after M. F. Hasler's observation *)
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PROG
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(PARI) A002072(n, a=[1, 8, 80, 4374, 9800, 123200, 336140, 11859210, 11859210, 177182720, 1611308699, 3463199999, 63927525375, 421138799639, 1109496723125, 1453579866024])=a[n] \\ "practical" solution for use in other sequences, easily extended to more values. - M. F. Hasler, Jan 16 2015
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CROSSREFS
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Cf. A002071, A003032, A003033, A122463, A145606, A175607.
Equals A117581(n) - 1.
Sequence in context: A347768 A240325 A145606 * A193943 A067449 A302974
Adjacent sequences: A002069 A002070 A002071 * A002073 A002074 A002075
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Don Reble, Jan 11 2005
a(18)-a(26) from Fred Schneider, Sep 09 2006
Corrected and extended by Andrey V. Kulsha, Aug 10 2011, according to Jim White's computations.
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STATUS
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approved
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