login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A109395 Denominator of phi(n)/n = Prod_{p|n} (1-1/p); phi(n)=A000010(n), the Euler totient function. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms 1..1000 from T. D. Noe)

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = n/gcd(n, phi(n)) = n/A009195(n).

From Antti Karttunen, Feb 09 2019: (Start)

a(n) = denominator of A173557(n)/A007947(n).

a(2^n) = 2 for all n >= 1.

(End)

From Amiram Eldar, Jul 31 2020: (Start)

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)

EXAMPLE

a(10) = 10/gcd(10,phi(10)) = 10/gcd(10,4) = 10/2 = 5.

MATHEMATICA

Table[Denominator[EulerPhi[n]/n], {n, 81}] (* Alonso del Arte, Sep 03 2011 *)

PROG

(PARI) a(n)=n/gcd(n, eulerphi(n)) \\ Charles R Greathouse IV, Feb 20 2013

(PARI)

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947

A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557

A109395(n) = denominator(A173557(n)/A007947(n)); \\ Antti Karttunen, Feb 09 2019

CROSSREFS

Cf. A076512 for the numerator.

Cf. A000010, A009195, A054741, A059956, A082695, A318304, A318305, A323170.

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).

Sequence in context: A323616 A102095 A331959 * A145254 A163457 A285708

Adjacent sequences:  A109392 A109393 A109394 * A109396 A109397 A109398

KEYWORD

nonn,frac,changed

AUTHOR

Franz Vrabec, Aug 26 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 05:31 EST 2020. Contains 338781 sequences. (Running on oeis4.)