cp's OEIS Frontend

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

%I A187731 #44 Jul 28 2020 11:14:23
%S A187731 1,2,3,5,7,11,13,15,17,19,23,29,31,37,41,43,47,51,53,59,61,67,71,73,
%T A187731 79,83,85,89,91,97,101,103,107,109,113,127,131,133,137,139,149,151,
%U A187731 157,163,167,173,179,181,191,193,197,199,211,223,227,229,233,239,241,247,251,255,257,259
%N A187731 Numbers n such that rad(phi(n)) divides n-1.
%C A187731 Subsequence of A003277 (cyclic numbers).
%C A187731 Let L(x) = exp(log x log log log x/log log x). McNew shows that there are at most x/L(x)^(1+o(1)) members of this sequence up to x. - _Charles R Greathouse IV_, Oct 08 2012
%C A187731 Contains all primes A000040 and all Carmichael numbers A002997. - _Jeppe Stig Nielsen_, Jul 27 2020
%H A187731 Amiram Eldar, <a href="/A187731/b187731.txt">Table of n, a(n) for n = 1..10000</a>
%H A187731 José María Grau and Antonio M. Oller-Marcén, <a href="http://arxiv.org/abs/1012.2337">On k-Lehmer numbers</a>, Integers, 12(2012), #A37
%H A187731 Nathan McNew, <a href="http://arxiv.org/abs/1210.2001">Radically weakening the Lehmer and Carmichael conditions</a> (2012)
%e A187731 15 is in the sequence because phi(15)=8, rad(8)=2 and 2 divides 15-1.
%t A187731 rad[n_]:=Times@@Transpose[FactorInteger[n]][[1]]; Select[Range[1000], Mod[#-1,rad[EulerPhi[#]]]==0&]
%o A187731 (PARI) rad(n)=n=factor(n);prod(i=1,#n[,1],n[i,1]);
%o A187731 for(n=1,1e4,if((n-1)%rad(eulerphi(n))==0,print1(n", "))) \\ _Charles R Greathouse IV_, Jul 04 2011
%o A187731 (PARI) is(n)=my(p=eulerphi(n), g=n); n--; while((g=gcd(p, g))>1, p/=g); p==1 && n \\ _Charles R Greathouse IV_, Mar 03 2014
%Y A187731 Cf. A000010, A002997 (Carmichael numbers), A003277 (cyclic numbers), A007947, A080400.
%K A187731 nonn
%O A187731 1,2
%A A187731 _José María Grau Ribas_, Mar 13 2011