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A009195
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a(n) = gcd(n, phi(n)).
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53
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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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refs;
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history;
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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a(n) = gcd(n, A051953(n)). - Labos Elemer
a(n) = n / A109395(n). - Antti Karttunen, May 04 2017 (corrected also typo in above formula).
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MAPLE
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a009195 := n -> igcd(i, numtheory[phi](n));
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MATHEMATICA
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Table[GCD[n, EulerPhi[n]], {n, 100}] (* Harvey P. Dale, Aug 11 2011 *)
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PROG
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(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
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CROSSREFS
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Cf. A000010, A003277, A050399, A051953, A061303, A109395, A285711.
Sequence in context: A200219 A270120 A325567 * A072994 A332741 A052126
Adjacent sequences: A009192 A009193 A009194 * A009196 A009197 A009198
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KEYWORD
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nonn,easy,nice
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AUTHOR
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David W. Wilson
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STATUS
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approved
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