A318304 -id:A318304 - cp's OEIS frontend

A109395 Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.

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

Offset: 1

Franz Vrabec, Aug 26 2005

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

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

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

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).

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

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

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

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)

A318305 a(n) = Product_{primes p dividing n} p - Product_{primes p dividing n} (p-1).

Original entry on oeis.org

0, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 4, 1, 8, 7, 1, 1, 4, 1, 6, 9, 12, 1, 4, 1, 14, 1, 8, 1, 22, 1, 1, 13, 18, 11, 4, 1, 20, 15, 6, 1, 30, 1, 12, 7, 24, 1, 4, 1, 6, 19, 14, 1, 4, 15, 8, 21, 30, 1, 22, 1, 32, 9, 1, 17, 46, 1, 18, 25, 46, 1, 4, 1, 38, 7, 20, 17, 54, 1, 6, 1, 42, 1, 30, 21, 44, 31, 12, 1, 22, 19, 24, 33, 48, 23, 4, 1, 8

Offset: 1

Antti Karttunen, Aug 26 2018

nonn

			For n = 45 = 3^2 * 5, the prime factors are 3 and 5, thus a(45) = (3*5) - (2*4) = 15 - 8 = 7.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065463 - A307868 = 0.232761... . - _Amiram Eldar_, Dec 07 2023

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A003557, A007947, A051953, A083254, A173557, A318304.

Cf. A065463, A307868.

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557
A051953(n) = (n - eulerphi(n));
A318305(n) = A051953(n)/A003557(n);

A007947(n) = factorback(factorint(n)[, 1]); \\ From A007947
A173557(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ From A173557
A318305(n) = (A007947(n) - A173557(n));

a(n) = A051953(n)/A003557(n) = A007947(n) - A173557(n) = A173557(n) - A318304(n).

Corrected the notation in the definition - Antti Karttunen, Feb 03 2024

cp's OEIS Frontend

A109395 Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.

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A318305 a(n) = Product_{primes p dividing n} p - Product_{primes p dividing n} (p-1).

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