

A129759


For the Landau function L(n), A000793, this sequence gives the largest prime which is a factor of L(n).


2



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, 13, 11, 11, 11, 11, 13, 13, 11, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 17, 17, 17, 17, 17, 19, 19, 17, 17, 17, 17, 19, 19, 17, 17, 19, 19, 19, 19, 19, 19, 17, 19
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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. 407410.
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. 665678. [alternate link]
JeanLouis Nicolas, Ordre maximal d'un élément du groupe S_n des permutations et 'highly composite numbers', Bull. Soc. Math. France 97 (1969), 129191.
Eric Weisstein's World of Mathematics, Landau's Function


FORMULA

a(n) = A006530(A000793(n)).  R. J. Mathar, May 17 2007


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]];
Array[a, 100] (* JeanFrançois Alcover, Feb 19 2020, after Alois P. Heinz in A000793 *)


CROSSREFS

Cf. A006530, A000793, A128305.
Sequence in context: A102351 A078173 A248005 * A137467 A261461 A078627
Adjacent sequences: A129756 A129757 A129758 * A129760 A129761 A129762


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



