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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

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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