OFFSET
1,2
COMMENTS
This function is not monotone increasing, for example a(33) = 13 while a(34) = 11.
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
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..10000
Jon Grantham, The largest prime divisor of the maximal order of an element of S_n, Math. Comp. 64:209 (1995), pp. 407-410.
J. P. Massias, J. L. Nicolas and G. Robin, Effective bounds for the maximal order of an element in the symmetric group, Math. Comp. 53:188 (1989), pp. 665-678. [alternate link]
Jean-Louis Nicolas, Ordre maximal d'un élément du groupe S_n des permutations et 'highly composite numbers', Bull. Soc. Math. France 97 (1969), 129-191.
Eric Weisstein's World of Mathematics, Landau's Function
FORMULA
EXAMPLE
L(29) = 2520, whose largest prime factor is 7. So a(29) = 7.
MATHEMATICA
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}]]]];
g[n_] := b[n, If[n<8, 3, PrimePi[Ceiling[1.328*Sqrt[n*Log[n] // Floor]]]]];
a[n_] := FactorInteger[g[n]][[-1, 1]];
CROSSREFS
KEYWORD
nonn,look
AUTHOR
Anthony C Robin, May 15 2007
EXTENSIONS
More terms from Klaus Brockhaus and R. J. Mathar, May 16 2007
Corrected a(66) by Alois P. Heinz, Feb 16 2013
STATUS
approved