A129759 For the Landau function L(n), A000793, this sequence gives the largest prime which is a factor of L(n).

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

Offset: 1

Author

Anthony C Robin, May 15 2007

Keywords

• nonn
• look

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

Examples

			L(29) = 2520, whose largest prime factor is 7. So a(29) = 7.
		

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

Crossrefs

Cf. A006530, A000793, A128305.

Programs

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] (* Jean-François Alcover, Feb 19 2020, after Alois P. Heinz in A000793 *)

Formula

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

Extensions

More terms from Klaus Brockhaus and R. J. Mathar, May 16 2007

Corrected a(66) by Alois P. Heinz, Feb 16 2013