A050384 - OEIS

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

COMMENTS

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

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

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

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 *)

PROG

(SageMath)

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