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A049559
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a(n) = gcd(n - 1, phi(n)).
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21
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1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 1, 2, 1, 16, 1, 18, 1, 4, 1, 22, 1, 4, 1, 2, 3, 28, 1, 30, 1, 4, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 46, 1, 6, 1, 2, 3, 52, 1, 2, 1, 4, 1, 58, 1, 60, 1, 2, 1, 16, 5, 66, 1, 4, 3, 70, 1, 72, 1, 2, 3, 4, 1, 78, 1, 2, 1, 82, 1, 4, 1, 2, 1, 88, 1, 18, 1, 4
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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For prime n, a(n) = n - 1. Question: do nonprimes exist with this property?
Answer: No. If n is composite then a(n) < n - 1. - Charles R Greathouse IV, Dec 09 2013
Lehmer's totient problem (1932): are there composite numbers n such that a(n) = phi(n)? - Thomas Ordowski, Nov 08 2015
a(n) = 1 for n in A209211. - Robert Israel, Nov 09 2015
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, B37.
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LINKS
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Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Lehmer's Totient Problem
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FORMULA
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a(p^m) = a(p) = p - 1 for prime p and m > 0. - Thomas Ordowski, Dec 10 2013
From Antti Karttunen, Sep 09 2018: (Start)
a(n) = A000010(n) / A160595(n) = A063994(n) / A318829(n).
a(n) = n - A318827(n) = A000010(n) - A318830(n).
(End)
a(2^n) = 1. a(p^n) = p - 1 for all
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EXAMPLE
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a(9) = 2 because phi(9) = 6 and gcd(8, 6) = 2.
a(10) = 1 because phi(10) = 4 and gcd(9, 4) = 1.
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MAPLE
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seq(igcd(n-1, numtheory:-phi(n)), n=1..100); # Robert Israel, Nov 09 2015
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MATHEMATICA
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Table[GCD[n - 1, EulerPhi[n]], {n, 93}] (* Michael De Vlieger, Nov 09 2015 *)
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PROG
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(PARI) a(n)=gcd(eulerphi(n), n-1) \\ Charles R Greathouse IV, Dec 09 2013
(Python)
from sympy import totient, gcd
print[gcd(totient(n), n - 1) for n in range(1, 101)] # Indranil Ghosh, Mar 27 2017
(MAGMA) [Gcd(n-1, EulerPhi(n)): n in [1..80]]; // Vincenzo Librandi, Oct 13 2018
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CROSSREFS
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Cf. A000010, A009195, A058515, A063994, A160595, A209211, A318827, A318829, A318830.
Sequence in context: A060680 A057237 A187730 * A063994 A268336 A295127
Adjacent sequences: A049556 A049557 A049558 * A049560 A049561 A049562
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KEYWORD
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nonn
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AUTHOR
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Labos Elemer, Dec 28 2000
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STATUS
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approved
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