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).

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

Table of n, a(n) for n=1..22.

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

Cf. A000010, A002322, A066605.

Sequence in context: A102476 A348418 A302109 * A220486 A180380 A057420

Adjacent sequences: A328269 A328270 A328271 * A328273 A328274 A328275

Keyword

nonn,more,hard

Author

Amiram Eldar, Oct 10 2019

Extensions

a(21)-a(22) from Lucas A. Brown, Feb 28 2024

Status

approved