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 A002072 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) 22
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 REFERENCES 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). LINKS 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 FORMULA a(n) < 10^n/n except for n=4. (Conjectured, from experimental data.) - M. F. Hasler, Jan 16 2015 EXAMPLE 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. MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A002071, A003032, A003033, A122463, A145606, A175607. Equals A117581(n) - 1. Sequence in context: A271443 A240325 A145606 * A193943 A067449 A302974 Adjacent sequences:  A002069 A002070 A002071 * A002073 A002074 A002075 KEYWORD nonn,nice AUTHOR EXTENSIONS 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 STATUS approved

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Last modified October 19 13:01 EDT 2019. Contains 328222 sequences. (Running on oeis4.)