OFFSET
1,4
COMMENTS
The inequality gcd(n, phi(n)) <= 2n exp(-sqrt(log 2 log n)) holds for all squarefree n >= 1 (Erdős, Luca, and Pomerance).
Erdős shows that for almost all n, a(n) ~ log log log log n. - Charles R Greathouse IV, Nov 23 2011
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), pp. 75-78.
Paul Erdős, Florian Luca, Carl Pomerance, On the proportion of numbers coprime to a given integer, in Anatomy of Integers, pp. 47-64, J.-M. De Koninck, A. Granville, F. Luca (editors), AMS, 2008.
Joshua Stucky, The distribution of gcd(n,phi(n)), arXiv:2402.13997 [math.NT], 2024.
FORMULA
a(n) = gcd(n, A051953(n)). - Labos Elemer
a(n) = n / A109395(n). - Antti Karttunen, May 04 2017 (corrected also typo in above formula).
MAPLE
a009195 := n -> igcd(i, numtheory[phi](n));
MATHEMATICA
Table[GCD[n, EulerPhi[n]], {n, 100}] (* Harvey P. Dale, Aug 11 2011 *)
PROG
(PARI) a(n)=gcd(n, eulerphi(n)) \\ Charles R Greathouse IV, Nov 23 2011
(Haskell)
a009195 n = n `gcd` a000010 n -- Reinhard Zumkeller, Feb 27 2012
(Python)
def a009195(n):
from math import gcd
phi = lambda x: len([i for i in range(x) if gcd(x, i) == 1])
return gcd(n, phi(n))
# Edward Minnix III, Dec 05 2015
(Magma) [Gcd(n, EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Dec 17 2015
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved