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A238574 k-Lehmer numbers: composite integers n such that phi(n) | (n-1)^k. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Composite numbers in A187731.

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.

EXAMPLE

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

MATHEMATICA

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

PROG

(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

CROSSREFS

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

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

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

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

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

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

Sequence in context: A020214 A127643 A227129 * A238575 A020144 A235040

Adjacent sequences:  A238571 A238572 A238573 * A238575 A238576 A238577

KEYWORD

nonn

AUTHOR

José María Grau Ribas, Mar 01 2014

STATUS

approved

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Last modified October 22 09:56 EDT 2019. Contains 328315 sequences. (Running on oeis4.)