A009195 a(n) = gcd(n, phi(n)).

1, 1, 1, 2, 1, 2, 1, 4, 3, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 8, 5, 2, 9, 4, 1, 2, 1, 16, 1, 2, 1, 12, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 16, 7, 10, 1, 4, 1, 18, 5, 8, 3, 2, 1, 4, 1, 2, 9, 32, 1, 2, 1, 4, 1, 2, 1, 24, 1, 2, 5, 4, 1, 6, 1, 16, 27, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 4, 3, 2, 1, 32, 1, 14, 3, 20

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy
• nice

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.

Crossrefs

Cf. A000010, A003277, A050399, A051953, A061303, A109395, A285711.

Programs

Haskell

a009195 n = n `gcd` a000010 n  -- Reinhard Zumkeller, Feb 27 2012

Magma

[Gcd(n, EulerPhi(n)): n in [1..100]]; // Vincenzo Librandi, Dec 17 2015

Maple

a009195 := n -> igcd(i,numtheory[phi](n));

Mathematica

Table[GCD[n,EulerPhi[n]],{n,100}] (* Harvey P. Dale, Aug 11 2011 *)

PARI

a(n)=gcd(n,eulerphi(n)) \\ Charles R Greathouse IV, Nov 23 2011

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

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).