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

Author

Giovanni Resta, Jan 01 2014

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A187731, A173703, A234958.

Programs

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 *)

PARI

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