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

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 91, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 247, 251, 255, 257, 259

Offset: 1

Author

José María Grau Ribas, Mar 13 2011

Keywords

• nonn

Comments

Subsequence of A003277 (cyclic numbers).

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

Contains all primes A000040 and all Carmichael numbers A002997. - Jeppe Stig Nielsen, Jul 27 2020

Examples

			15 is in the sequence because phi(15)=8, rad(8)=2 and 2 divides 15-1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• 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 (2012)

Crossrefs

Cf. A000010, A002997 (Carmichael numbers), A003277 (cyclic numbers), A007947, A080400.

Programs

Mathematica

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

PARI

rad(n)=n=factor(n);prod(i=1,#n[,1],n[i,1]);
for(n=1,1e4,if((n-1)%rad(eulerphi(n))==0,print1(n", "))) \\ Charles R Greathouse IV, Jul 04 2011

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