login
A328272
Numbers k >= 3 where a record value of log(phi(k))/log(lambda(k)) is reached, where phi is the Euler totient function (A000010) and lambda is the Carmichael lambda function (A002322).
1
3, 8, 24, 7280, 13104, 21840, 32760, 65520, 2878785, 5117840, 6909084, 8636355, 19740240, 27636336, 46060560, 69090840, 138181680, 1703601900, 2271469200, 3407203800, 6814407600, 20174525280
OFFSET
1,1
COMMENTS
Banks et al. proved that the set {log(phi(k))/log(lambda(k)) | k >= 3} is dense in [1, oo). Therefore this sequence is infinite.
Subsequent terms exceed 10^11. - Lucas A. Brown, Feb 28 2024
LINKS
William D. Banks, Kevin Ford, Florian Luca, Francesco Pappalardi and Igor E. Shparlinski, Values of the Euler Function in Various Sequences, Monatshefte für Mathematik, Vol. 146, No. 1 (2005), pp 1-19, alternative link.
Lucas A. Brown, Python program.
EXAMPLE
For k < 8, phi(k) = lambda(k), and log(phi(k))/log(lambda(k))} = 1. For k = 8, phi(8) = 4 and lambda(8) = 2, so log(phi(8))/log(lambda(8)) = log(4)/log(2) = 2 is a record value, and hence 8 is in this sequence.
MATHEMATICA
r[n_] := Log[EulerPhi[n]]/Log[CarmichaelLambda[n]]; rm = 0; s = {}; Do[r1 = r[n]; If[r1 > rm, rm = r1; AppendTo[s, n]], {n, 3, 10^5}]; s
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Amiram Eldar, Oct 10 2019
EXTENSIONS
a(21)-a(22) from Lucas A. Brown, Feb 28 2024
STATUS
approved