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

%I A238574 #82 Oct 10 2020 04:06:26
%S A238574 15,51,85,91,133,247,255,259,435,451,481,511,561,595,679,703,763,771,
%T A238574 949,1105,1111,1141,1261,1285,1351,1387,1417,1615,1695,1729,1843,1891,
%U A238574 2047,2071,2091,2119,2431,2465,2509,2701,2761,2821,2955,3031,3097,3145,3277
%N A238574 k-Lehmer numbers: composite integers n such that phi(n) | (n-1)^k.
%C A238574 Composite numbers in A187731.
%C A238574 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
%H A238574 Charles R Greathouse IV, <a href="/A238574/b238574.txt">Table of n, a(n) for n = 1..10000</a>
%H A238574 J. M. Grau and A. M. Oller-Marcén, <a href="http://www.emis.de/journals/INTEGERS/papers/m37/m37.Abstract.html">On k-Lehmer numbers</a>, Integers, 12(2012), #A37.
%H A238574 Max Lewis and Victor Scharaschkin, <a href="https://www.emis.de/journals/INTEGERS/papers/q80/q80.Abstract.html">k-Lehmer and k-Carmichael Numbers</a>, Integers, 16 (2016), #A80.
%H A238574 Nathan McNew, <a href="http://arxiv.org/abs/1210.2001">Radically weakening the Lehmer and Carmichael conditions</a>, arXiv:1210.2001 [math.NT], 2012; International Journal of Number Theory 9 (2013), 1215-1224.
%H A238574 Nathan McNew, <a href="http://libarchive.dartmouth.edu/cdm/ref/collection/dcdis/id/187971">Multiplicative problems in combinatorial number theory</a>, Thesis, 2015.
%H A238574 Nathan McNew and Thomas Wright, <a href="http://www.nathanmcnew.com/radimichael.pdf">Infinitude of k-Lehmer numbers which are not Carmichael</a>, Int. J. Number Theory V.12(7), pp. 1863-1869, (2016).
%H A238574 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/Lehmer_number/">k-Lehmer numbers</a>.
%e A238574 2^3*3^2 = 72 = phi(91) divides (91-1)^3 = (2*3^2*5)^3 implies 91 is a 3-Lehmer number.
%t A238574 rad[n_]:=Times@@Transpose[FactorInteger[n]][[1]]; Select[1+Range[1000], !PrimeQ[#]&&Mod[#-1, rad[EulerPhi[#]]]==0&]
%o A238574 (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
%Y A238574 Cf. A187731 (numbers n such that rad(phi(n)) divides n-1).
%Y A238574 Cf. A173703 (2-Lehmer numbers; i.e., phi(n) divides (n-1)^2).
%Y A238574 Cf. A234936 (smallest composite n-Lehmer number which is not an (n-1)-Lehmer number).
%Y A238574 Cf. A207080 (minimum Carmichael number which is not an n-Lehmer number).
%Y A238574 Cf. A234958 (number of k-Lehmer numbers up to 10^n).
%Y A238574 Cf. A238575 (k-Lehmer numbers with two prime factors).
%Y A238574 Cf. A002997, A003277.
%K A238574 nonn
%O A238574 1,1
%A A238574 _José María Grau Ribas_, Mar 01 2014