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%I M4560 N1942 #72 Oct 05 2020 10:56:20
%S 1,8,80,4374,9800,123200,336140,11859210,11859210,177182720,
%T 1611308699,3463199999,63927525375,421138799639,1109496723125,
%U 1453579866024,20628591204480,31887350832896,31887350832896,119089041053696,2286831727304144,9591468737351909375,9591468737351909375,9591468737351909375,9591468737351909375,9591468737351909375,19316158377073923834000
%N a(n) = smallest number m such that for all k > m, either k or k+1 has a prime factor > prime(n).
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H E. F. Ecklund and R. B. Eggleton, <a href="http://www.jstor.org/stable/2317422">Prime factors of consecutive integers</a>, Amer. Math. Monthly, 79 (1972), 1082-1089.
%H D. H. Lehmer, <a href="http://projecteuclid.org/euclid.ijm/1256067456">On a problem of Størmer</a>, Ill. J. Math., 8 (1964), 57-79.
%H Don Reble, <a href="/A002072/a002072.py.txt">Python program</a>
%H Jim White, <a href="https://11011110.github.io/blog/2007/03/23/smooth-pairs.html">Results to P = 127</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Stormer%27s_theorem">Størmer's theorem</a>
%F a(n) < 10^n/n except for n=4. (Conjectured, from experimental data.) - _M. F. Hasler_, Jan 16 2015
%e 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.)
%e a(2) = 8 since for any larger k, we cannot have k and k+1 both 3-smooth (cf. A003586).
%e 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.
%t 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 *)
%o (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
%Y Cf. A002071, A003032, A003033, A122463, A145606, A175607.
%Y Equals A117581(n) - 1.
%K nonn,nice
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _Don Reble_, Jan 11 2005
%E a(18)-a(26) from _Fred Schneider_, Sep 09 2006
%E Corrected and extended by _Andrey V. Kulsha_, Aug 10 2011, according to Jim White's computations.
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