A238574 k-Lehmer numbers: composite integers n such that phi(n) | (n-1)^k.

15, 51, 85, 91, 133, 247, 255, 259, 435, 451, 481, 511, 561, 595, 679, 703, 763, 771, 949, 1105, 1111, 1141, 1261, 1285, 1351, 1387, 1417, 1615, 1695, 1729, 1843, 1891, 2047, 2071, 2091, 2119, 2431, 2465, 2509, 2701, 2761, 2821, 2955, 3031, 3097, 3145, 3277

Offset: 1

Author

José María Grau Ribas, Mar 01 2014

Keywords

• nonn

Comments

Composite numbers in A187731.

J. M. Grau and A. M. Oller-Marcén showed that all terms of this sequence are terms of A003277 (cyclic numbers) and this sequence contains all terms of A002997 (Carmichael numbers). - Tomohiro Yamada, Sep 28 2020

Examples

			2^3*3^2 = 72 = phi(91) divides (91-1)^3 = (2*3^2*5)^3 implies 91 is a 3-Lehmer number.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• J. M. Grau and A. M. Oller-Marcén, On k-Lehmer numbers, Integers, 12(2012), #A37.
• Max Lewis and Victor Scharaschkin, k-Lehmer and k-Carmichael Numbers, Integers, 16 (2016), #A80.
• Nathan McNew, Radically weakening the Lehmer and Carmichael conditions, arXiv:1210.2001 [math.NT], 2012; International Journal of Number Theory 9 (2013), 1215-1224.
• Nathan McNew, Multiplicative problems in combinatorial number theory, Thesis, 2015.
• Nathan McNew and Thomas Wright, Infinitude of k-Lehmer numbers which are not Carmichael, Int. J. Number Theory V.12(7), pp. 1863-1869, (2016).
• Giovanni Resta, k-Lehmer numbers.

Crossrefs

Cf. A187731 (numbers n such that rad(phi(n)) divides n-1).

Cf. A173703 (2-Lehmer numbers; i.e., phi(n) divides (n-1)^2).

Cf. A234936 (smallest composite n-Lehmer number which is not an (n-1)-Lehmer number).

Cf. A207080 (minimum Carmichael number which is not an n-Lehmer number).

Cf. A234958 (number of k-Lehmer numbers up to 10^n).

Cf. A238575 (k-Lehmer numbers with two prime factors).

Cf. A002997, A003277.

Programs

Mathematica

rad[n_]:=Times@@Transpose[FactorInteger[n]][[1]]; Select[1+Range[1000], !PrimeQ[#]&&Mod[#-1, rad[EulerPhi[#]]]==0&]

PARI

is(n)=my(p=eulerphi(n),g=n); if(isprime(n),return(0),n--); while((g=gcd(p,g))>1, p/=g); p==1 && n \\ Charles R Greathouse IV, Mar 03 2014