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

1, 15, 33, 35, 51, 65, 69, 77, 85, 87, 91, 95, 115, 119, 123, 133, 141, 143, 145, 159, 161, 177, 185, 187, 209, 213, 215, 217, 221, 235, 247, 249, 255, 259, 265, 267, 287, 295, 299, 303, 319, 321, 323, 329, 335, 339, 341, 345, 365, 371, 377, 391, 393, 395, 403

Offset: 1

Author

Christian G. Bower, Nov 15 1999

Keywords

• nonn

Comments

Also nonprimes n such that there is only one group of order n, i.e., A000001(n) = 1.

Intersection of A018252 and A003277.

Also numbers n such that n and A051953(n) are relatively prime. - Labos Elemer

Apart from the first term, this is a subsequence of A024556. - Charles R Greathouse IV, Apr 15 2015

Every Carmichael number and each of its nonprime divisors is in this sequence. - Emmanuel Vantieghem, Apr 20 2015

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See pp. 2, 6.
• Peter Luschny, Strong coprimality, 2011.

Crossrefs

If the primes are included we get A003277. Cf. A000001, A000010 (phi), A181830, A181837.

Programs

Maple

isStrongPrimeTo := (n, k) -> (igcd(n, k) = 1) and not (irem(n-1, k) = 0):
isStrongCyclic := n -> isStrongPrimeTo(n, numtheory:-phi(n)):
[1, op(select(isStrongCyclic, [$(2..404)]))]; # Peter Luschny, Dec 13 2021

Mathematica

Select[Range[450], !PrimeQ[#] && GCD[#, EulerPhi[#]] == 1&] (* Harvey P. Dale, Jan 31 2011 *)

PARI

is(n)=!isprime(n) && gcd(eulerphi(n),n)==1 \\ Charles R Greathouse IV, Apr 15 2015

Sage

def isStrongPrimeTo(n, m): return gcd(n, m) == 1 and not m.divides(n-1)
def isStrongCyclic(n): return isStrongPrimeTo(n, euler_phi(n))
[1] + [n for n in (1..403) if isStrongCyclic(n)] # Peter Luschny, Nov 14 2018