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A076512 Denominator of cototient(n)/totient(n). 19

%I #42 Sep 08 2022 08:45:07

%S 1,1,2,1,4,1,6,1,2,2,10,1,12,3,8,1,16,1,18,2,4,5,22,1,4,6,2,3,28,4,30,

%T 1,20,8,24,1,36,9,8,2,40,2,42,5,8,11,46,1,6,2,32,6,52,1,8,3,12,14,58,

%U 4,60,15,4,1,48,10,66,8,44,12,70,1,72,18,8,9,60,4,78,2,2,20,82,2,64,21

%N Denominator of cototient(n)/totient(n).

%C a(n)=1 iff n=A007694(k) for some k.

%C Numerator of phi(n)/n=Prod_{p|n} (1-1/p). - _Franz Vrabec_, Aug 26 2005

%C From _Wolfdieter Lang_, May 12 2011: (Start)

%C For n>=2, a(n)/A109395(n) = sum(((-1)^r)*sigma_r,r=0..M(n)) with the elementary symmetric functions (polynomials) sigma_r of the indeterminates {1/p_1,...,1/p_M(n)} if n = prod((p_j)^e(j),j=1..M(n)) where M(n)=A001221(n) and sigma_0=1.

%C This follows by expanding the above given product for phi(n)/n.

%C The n-th member of this rational sequence 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5,... is also (2/n^2)*sum(k,with 1<=k<n and gcd(k,n)=1), n>=2.

%C Therefore, this scaled sum depends only on the distinct prime factors of n.

%C See also A023896. Proof via PIE (principle of inclusion and exclusion). (End)

%C In the sequence of rationals r(n)=eulerphi(n)/n: 1, 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5, 10/11, 1/3, ... one can observe that new values are obtained for squarefree indices (A005117); while for a nonsquarefree number n (A013929), r(n) = r(A007947(n)), where A007947(n) is the squarefree kernel of n. - _Michel Marcus_, Jul 04 2015

%H T. D. Noe, <a href="/A076512/b076512.txt">Table of n, a(n) for n=1..1000</a>

%F a(n) = A000010(n)/A009195(n).

%t Table[Denominator[(n - EulerPhi[n])/EulerPhi[n]], {n, 80}] (* _Alonso del Arte_, May 12 2011 *)

%o (PARI) vector(80, n, numerator(eulerphi(n)/n)) \\ _Michel Marcus_, Jul 04 2015

%o (Magma) [Numerator(EulerPhi(n)/n): n in [1..100]]; // _Vincenzo Librandi_, Jul 04 2015

%Y Cf. A076511 (numerator of cototient(n)/totient(n)), A051953.

%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 1,3

%A _Reinhard Zumkeller_, Oct 15 2002

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