cp's OEIS Frontend

A050384 Nonprimes such that n and phi(n) are relatively prime.

%I A050384 #44 Aug 15 2025 10:33:48
%S A050384 1,15,33,35,51,65,69,77,85,87,91,95,115,119,123,133,141,143,145,159,
%T A050384 161,177,185,187,209,213,215,217,221,235,247,249,255,259,265,267,287,
%U A050384 295,299,303,319,321,323,329,335,339,341,345,365,371,377,391,393,395,403
%N A050384 Nonprimes such that n and phi(n) are relatively prime.
%C A050384 Also nonprimes n such that there is only one group of order n, i.e., A000001(n) = 1.
%C A050384 Intersection of A018252 and A003277.
%C A050384 Also numbers n such that n and A051953(n) are relatively prime. - _Labos Elemer_
%C A050384 Apart from the first term, this is a subsequence of A024556. - _Charles R Greathouse IV_, Apr 15 2015
%C A050384 Every Carmichael number and each of its nonprime divisors is in this sequence. - _Emmanuel Vantieghem_, Apr 20 2015
%C A050384 An alternative definition (excluding the 1): k is strongly prime to n <=> k is prime to n and k does not divide n - 1 (cf. A181830). n is cyclic if n is prime to phi(n). n is strongly cyclic if phi(n) is strongly prime to n. The a(n) are the strongly cyclic numbers apart from a(1). - _Peter Luschny_, Nov 14 2018
%H A050384 T. D. Noe, <a href="/A050384/b050384.txt">Table of n, a(n) for n = 1..10000</a>
%H A050384 Joel E. Cohen, <a href="https://arxiv.org/abs/2508.08335">Conjectures about Primes and Cyclic Numbers</a>, arXiv:2508.08335 [math.NT], 2025. See pp. 2, 6.
%H A050384 Peter Luschny, <a href="/wiki/User:Peter_Luschny/StrongCoprimality">Strong coprimality</a>, 2011.
%p A050384 isStrongPrimeTo := (n, k) -> (igcd(n, k) = 1) and not (irem(n-1, k) = 0):
%p A050384 isStrongCyclic := n -> isStrongPrimeTo(n, numtheory:-phi(n)):
%p A050384 [1, op(select(isStrongCyclic, [$(2..404)]))]; # _Peter Luschny_, Dec 13 2021
%t A050384 Select[Range[450], !PrimeQ[#] && GCD[#, EulerPhi[#]] == 1&] (* _Harvey P. Dale_, Jan 31 2011 *)
%o A050384 (PARI) is(n)=!isprime(n) && gcd(eulerphi(n),n)==1 \\ _Charles R Greathouse IV_, Apr 15 2015
%o A050384 (Sage)
%o A050384 def isStrongPrimeTo(n, m): return gcd(n, m) == 1 and not m.divides(n-1)
%o A050384 def isStrongCyclic(n): return isStrongPrimeTo(n, euler_phi(n))
%o A050384 [1] + [n for n in (1..403) if isStrongCyclic(n)] # _Peter Luschny_, Nov 14 2018
%Y A050384 If the primes are included we get A003277. Cf. A000001, A000010 (phi), A181830, A181837.
%K A050384 nonn
%O A050384 1,2
%A A050384 _Christian G. Bower_, Nov 15 1999