A234936 a(n) is the smallest composite n-Lehmer number.

561, 15, 451, 51, 679, 255, 2091, 771, 43435, 3855, 31611, 13107, 272163, 65535, 494211, 196611, 2089011, 983055, 8061051, 3342387, 31580931, 16711935, 126027651, 50529027, 756493591, 252645135, 4446487299, 858993459, 8053383171, 4294967295, 32212942851

Offset

2,1

Comments

A number n is a k-Lehmer number if there exists a k such that phi(n) divides (n-1)^k, but not (n-1)^(k-1). The existence of a composite 1-Lehmer number is deemed improbable.

Links

Giovanni Resta, Table of n, a(n) for n = 2..36

José María Grau and Antonio M. Oller-Marcén, On k-Lehmer numbers, arXiv:1012.2337 [math.NT], 2010-2012.

José María Grau and Antonio M. Oller-Marcén, On k-Lehmer numbers, Integers, 12(2012), #A37.

Nathan McNew, Radically weakening the Lehmer and Carmichael conditions, arXiv:1210.2001 [math.NT], 2012; International Journal of Number Theory 9 (2013), 1215-1224.

Example

a(3) = 15 because 15 is the smallest n such that phi(n) divides (n-1)^3 and does not divide (n-1)^2, i.e., it is the smallest 3-Lehmer number.

Mathematica

a[n_] := a[n] = For[k = 2, True, k++, If[CompositeQ[k], phi = EulerPhi[k]; If[Divisible[(k-1)^n, phi], If[!Divisible[(k-1)^(n-1), phi], Return[k] ]]]]; Table[Print[n, " ", a[n]]; a[n], {n, 2, 20}] (* Jean-François Alcover, Jan 26 2019 *)

Prog

(PARI) a(n) = {x = 2; while (!(!((x-1)^n % eulerphi(x)) && ((x-1)^(n-1) % eulerphi(x))), x++); x; } \\ Michel Marcus, Jan 26 2014

Crossrefs

Cf. A187731, A173703, A234958.

Sequence in context: A104590 A259211 A265261 * A274443 A247906 A278338

Adjacent sequences: A234933 A234934 A234935 * A234937 A234938 A234939

Keyword

nonn

Author

Giovanni Resta, Jan 01 2014

Status

approved