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%I #29 Feb 19 2020 09:55:16
%S 1,2,3,2,3,3,3,5,5,5,5,5,5,7,7,7,7,7,7,7,7,7,7,7,7,7,11,11,7,11,11,13,
%T 13,11,11,11,11,13,13,11,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,
%U 13,17,17,17,17,17,19,19,17,17,17,17,19,19,17,17,19,19,19,19,19,19,17,19
%N For the Landau function L(n), A000793, this sequence gives the largest prime which is a factor of L(n).
%C This function is not monotone increasing, for example a(33) = 13 while a(34) = 11.
%C Nicolas showed that a(n) ~ sqrt(n log n) and Grantham showed that a(n) <= 1.328 sqrt(n log n) for n > 4. Massias, Nicolas, & Robin conjecture that a(n) <= 1.265... sqrt(n log n) in this range with equality at n = 215. - _Charles R Greathouse IV_, Jun 02 2014
%H Alois P. Heinz, <a href="/A129759/b129759.txt">Table of n, a(n) for n = 1..10000</a>
%H Jon Grantham, <a href="http://www.pseudoprime.com/maxord.html">The largest prime divisor of the maximal order of an element of S_n</a>, Math. Comp. 64:209 (1995), pp. 407-410.
%H J. P. Massias, J. L. Nicolas and G. Robin, <a href="http://math.univ-lyon1.fr/~nicolas/gdenMathComp.pdf">Effective bounds for the maximal order of an element in the symmetric group</a>, Math. Comp. 53:188 (1989), pp. 665-678. [<a href="https://doi.org/10.1090/S0025-5718-1989-0979940-4">alternate link</a>]
%H Jean-Louis Nicolas, <a href="https://doi.org/10.24033/bsmf.1676">Ordre maximal d'un élément du groupe S_n des permutations et 'highly composite numbers'</a>, Bull. Soc. Math. France 97 (1969), 129-191.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LandausFunction.html">Landau's Function</a>
%F a(n) = A006530(A000793(n)). - _R. J. Mathar_, May 17 2007
%e L(29) = 2520, whose largest prime factor is 7. So a(29) = 7.
%t b[n_, i_] := b[n, i] = Module[{p}, p = If[i < 1, 1, Prime[i]]; If[n == 0 || i < 1, 1, Max[b[n, i - 1], Table[p^j*b[n - p^j, i - 1], {j, 1, Log[p, n] // Floor}]]]];
%t g[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]];
%t a[n_] := FactorInteger[g[n]][[-1, 1]];
%t Array[a, 100] (* _Jean-François Alcover_, Feb 19 2020, after _Alois P. Heinz_ in A000793 *)
%Y Cf. A006530, A000793, A128305.
%K nonn,look
%O 1,2
%A _Anthony C Robin_, May 15 2007
%E More terms from _Klaus Brockhaus_ and _R. J. Mathar_, May 16 2007
%E Corrected a(66) by _Alois P. Heinz_, Feb 16 2013
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