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A007431
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a(n) = Sum_{d|n} phi(d)*mu(n/d).
(Formerly M2197)
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33
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0, 1, 0, 1, 1, 3, 0, 5, 2, 4, 0, 9, 1, 11, 0, 3, 4, 15, 0, 17, 3, 5, 0, 21, 2, 16, 0, 12, 5, 27, 0, 29, 8, 9, 0, 15, 4, 35, 0, 11, 6, 39, 0, 41, 9, 12, 0, 45, 4, 36, 0, 15, 11, 51, 0, 27, 10, 17, 0, 57, 3, 59, 0, 20, 16, 33, 0, 65, 15, 21, 0, 69, 8, 71, 0, 16, 17, 45, 0, 77, 12, 36, 0, 81, 5, 45, 0
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OFFSET
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0,6
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COMMENTS
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Also Moebius transform applied twice to natural numbers.
Also number of complex primitive Dirichlet characters modulo n and Sum_{k=1..n} a(k) is asymptotic to (18/Pi^4)*n^2. - Steven Finch, Feb 16 2006
a(n) is the number of degree-psi(n) primitive Dirichlet characters mod n, where psi = A002322. Also, a(n) is the number of degree-(k*psi(n)) primitive Dirichlet characters mod n for all k >= 1.
a(n) is the maximum element in the n-th row of A354058 (or A354061). (End)
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Multiplicative with a(p) = p-2 and a(p^e) = (p-1)^2*p^(e-2) for e > 1. - Vladeta Jovovic, Jan 25 2002
Dirichlet g.f.: zeta(s-1)/zeta^2(s).
a(n) = Sum_{k=1..n} mu(gcd(n,k)) for n > 0. - Benoit Cloitre, Jun 14 2007
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} mu(n)*x^n/(1 - x^n)^2. - Ilya Gutkovskiy, Apr 25 2017
Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} phi(n)*x^n. - Mamuka Jibladze, Aug 09 2019
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
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EXAMPLE
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a(45) = 12: psi(45) = 12, there are 3 degree-12 primitive characters modulo 5 and 4 degree-12 primitive characters modulo 9, so a(45) = 3 * 4 = 12.
a(63) = 20: psi(63) = 6, there are 5 sextic primitive characters modulo 7 and 4 sextic primitive characters modulo 9, so a(63) = 5 * 4 = 20. (End)
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MAPLE
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with(numtheory); f:=n->add( phi(d)*mobius(n/d), d in divisors(n)); [seq(f(n), n=0..120)];
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MATHEMATICA
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Table[Sum[EulerPhi[d] MoebiusMu[n/d], {d, Divisors[n]}], {n, 0, 86}] (* Jean-François Alcover, Apr 04 2011 *)
Table[DirichletConvolve[MoebiusMu[n], EulerPhi[n], n, m], {m, 86}] (* Jan Mangaldan, Mar 15 2013 *)
f[p_, e_] := If[e == 1, p-2, p^e - 2*p^(e-1) + p^(e-2)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 23 2020 *)
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PROG
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(PARI) a(n)=if(n<1, 0, direuler(p=2, n, (1-X)^2/(1-p*X))[n]) \\ Ralf Stephan
(PARI) a(n) = sumdiv(n, d, moebius(d) * eulerphi(n/d) ); \\ Joerg Arndt, Apr 14 2013
(Haskell)
a007431 0 = 0
a007431 n = sum $ map (a008683 . gcd n) [1..n]
(Magma) [0] cat [&+[EulerPhi(d)*MoebiusMu(Floor(n/d)):d in Divisors(n)]:n in [1..90]]; // Marius A. Burtea, Aug 10 2019
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CROSSREFS
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KEYWORD
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nonn,nice,mult
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AUTHOR
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STATUS
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approved
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