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

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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, Oct 15 2002

Keywords

• nonn
• frac

Comments

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

Numerator of phi(n)/n=Prod_{p|n} (1-1/p). - Franz Vrabec, Aug 26 2005

From Wolfdieter Lang, May 12 2011: (Start)

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.

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

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=2.

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

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

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

Links

• T. D. Noe, Table of n, a(n) for n=1..1000

Crossrefs

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

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).

Programs

Magma

[Numerator(EulerPhi(n)/n): n in [1..100]]; // Vincenzo Librandi, Jul 04 2015

Mathematica

Table[Denominator[(n - EulerPhi[n])/EulerPhi[n]], {n, 80}] (* Alonso del Arte, May 12 2011 *)

PARI

vector(80, n, numerator(eulerphi(n)/n)) \\ Michel Marcus, Jul 04 2015

Formula

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