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%I #41 Jan 07 2021 01:20:44
%S 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,
%T 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,
%U 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
%N Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.
%C 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
%H Antti Karttunen, <a href="/A109395/b109395.txt">Table of n, a(n) for n = 1..16384</a> (terms 1..1000 from T. D. Noe)
%H Antti Karttunen, <a href="/A109395/a109395.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F a(n) = n/gcd(n, phi(n)) = n/A009195(n).
%F From _Antti Karttunen_, Feb 09 2019: (Start)
%F a(n) = denominator of A173557(n)/A007947(n).
%F a(2^n) = 2 for all n >= 1.
%F (End)
%F From _Amiram Eldar_, Jul 31 2020: (Start)
%F Asymptotic mean of phi(n)/n: lim_{m->oo} (1/m) * Sum_{n=1..m} A076512(n)/a(n) = 6/Pi^2 (A059956).
%F 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)
%e a(10) = 10/gcd(10,phi(10)) = 10/gcd(10,4) = 10/2 = 5.
%t Table[Denominator[EulerPhi[n]/n], {n, 81}] (* _Alonso del Arte_, Sep 03 2011 *)
%o (PARI) a(n)=n/gcd(n,eulerphi(n)) \\ _Charles R Greathouse IV_, Feb 20 2013
%o (PARI)
%o A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
%o A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
%o A109395(n) = denominator(A173557(n)/A007947(n)); \\ _Antti Karttunen_, Feb 09 2019
%Y Cf. A076512 for the numerator.
%Y Cf. A000010, A009195, A054741, A059956, A082695, A318304, A318305, A323170.
%Y 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).
%K nonn,frac
%O 1,2
%A _Franz Vrabec_, Aug 26 2005
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