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A234936
a(n) is the smallest composite n-Lehmer number.
5
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
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
KEYWORD
nonn
AUTHOR
Giovanni Resta, Jan 01 2014
STATUS
approved