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A109395
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Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.
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23
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1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 15, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 2, 33, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 11, 7, 19, 29, 59, 15, 61, 31, 7, 2, 65, 33, 67, 17, 69, 35, 71, 3, 73, 37, 15, 19, 77, 13, 79, 5, 3
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OFFSET
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1,2
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COMMENTS
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a(n)=2 iff n=2^k (k>0); otherwise a(n) is odd. If p is prime, a(p)=p; the converse is false, e.g.: a(15)=15. It is remarkable that this sequence often coincides with A006530, the largest prime P dividing n. Theorem: a(n)=P if and only if for every prime p < P in n there is some prime q in n with p|(q-1). - Franz Vrabec, Aug 30 2005
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LINKS
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FORMULA
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a(n) = n/gcd(n, phi(n)) = n/A009195(n).
a(2^n) = 2 for all n >= 1.
(End)
Asymptotic mean of phi(n)/n: lim_{m->oo} (1/m) * Sum_{n=1..m} A076512(n)/a(n) = 6/Pi^2 (A059956).
Asymptotic mean of n/phi(n): lim_{m->oo} (1/m) * Sum_{n=1..m} a(n)/A076512(n) = zeta(2)*zeta(3)/zeta(6) (A082695). (End)
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EXAMPLE
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a(10) = 10/gcd(10,phi(10)) = 10/gcd(10,4) = 10/2 = 5.
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MATHEMATICA
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Table[Denominator[EulerPhi[n]/n], {n, 81}] (* Alonso del Arte, Sep 03 2011 *)
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PROG
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(PARI)
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
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CROSSREFS
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Phi(m)/m = k: A000079 \ {1} (k=1/2), A033845 (k=1/3), A000244 \ {1} (k=2/3), A033846 (k=2/5), A000351 \ {1} (k=4/5), A033847 (k=3/7), A033850 (k=4/7), A000420 \ {1} (k=6/7), A033848 (k=5/11), A001020 \ {1} (k=10/11), A288162 (k=6/13), A001022 \ {1} (12/13), A143207 (k=4/15), A033849 (k=8/15), A033851 (k=24/35).
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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