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 A109395 Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function. 22
 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 AUTHOR Franz Vrabec, Aug 26 2005 STATUS approved

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Last modified November 30 19:14 EST 2022. Contains 358453 sequences. (Running on oeis4.)