A003277 - OEIS
A003277
Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.
(Formerly M0650)
87
1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 157, 159, 161, 163, 167, 173
COMMENTS
Except for a(2)=2, all the terms in the sequence are odd. This is because of the existence of a non-cyclic dihedral group of order 2n for each n>1. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 09 2001
n such that x^n == 1 (mod n) has no solution 2 <= x <= n. - Benoit Cloitre, May 10 2002
There is only one group (the cyclic group of order n) whose order is n. - Gerard P. Michon, Jan 08 2008 [This is a 1947 result of Tibor Szele. - Charles R Greathouse IV, Nov 23 2011]
Any divisor of a Carmichael number (A002997) must be odd and cyclic. Conversely, G. P. Michon conjectured (c. 1980) that any odd cyclic number has at least one Carmichael multiple (if the conjecture is true, each of them has infinitely many such multiples). In 2007, Michon & Crump produced explicit Carmichael multiples of all odd cyclic numbers below 10000 (see link, cf. A253595). - Gerard P. Michon, Jan 08 2008
Numbers n such that phi(n)^phi(n) == 1 (mod n). - Michel Lagneau, Nov 18 2012
Number m such that n^n == r (mod m) is solvable for any r. - David W. Wilson, Oct 01 2015
Squarefree terms of A056867 (see McCarthy link p. 592 and similar comment with "cubefree" in A051532). - Bernard Schott, Mar 24 2022
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Jonathan Pakianathan and Krishnan Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
FORMULA
n = p_1*p_2*...*p_k (for some k >= 0), where the p_i are distinct primes and no p_j-1 is divisible by any p_i.
MAPLE
select(t -> igcd(t, numtheory:-phi(t))=1, [$1..1000]); # Robert Israel, Jul 08 2015
PROG
(PARI) isA003277(n) = gcd(n, eulerphi(n))==1 \\ Michael B. Porter, Feb 21 2010
(Haskell)
import Data.List (elemIndices)
a003277 n = a003277_list !! (n-1)
a003277_list = map (+ 1) $ elemIndices 1 a009195_list
(Magma) [n: n in [1..200] | Gcd(n, EulerPhi(n)) eq 1]; // Vincenzo Librandi, Jul 09 2015
(SageMath) # Compare A050384.
def isPrimeTo(n, m): return gcd(n, m) == 1
def isCyclic(n): return isPrimeTo(n, euler_phi(n))
[n for n in (1..173) if isCyclic(n)] # Peter Luschny, Nov 14 2018