OFFSET
0,6
COMMENTS
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
Dirichlet convolution of phi(n) and mu(n). - Richard L. Ollerton, May 07 2021
From Jianing Song, May 21 2022: (Start)
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.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..20000 (terms 0..1000 from T. D. Noe)
H. Jager, On the number of Dirichlet characters with modulus not exceeding x, Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math. 35 (1973) 452-455.
Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory A50 (8(1)), 2008.
N. J. A. Sloane, Transforms
FORMULA
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
a(n) = Sum_{k=1..n} (phi(gcd(k,n)) * cos(2*Pi*k/n)). - Enrique Pérez Herrero, Jan 18 2013
a(n) = Sum_{d|n} tau_{-2}(d)*n/d = Sum_{d|n} tau_{-3}(d)*sigma_1(n/d), where tau_{-3} is A007428, tau_{-2} A007427 and sigma_1 A000203. - Enrique Pérez Herrero, Jan 19 2013
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_{k=1..n} a(k) ~ 18 * n^2 / Pi^4. - Vaclav Kotesovec, Nov 04 2018
Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} phi(n)*x^n. - Mamuka Jibladze, Aug 09 2019
Sum_{d|n} a(d) = phi(n) (A000010). - Amiram Eldar, Jun 23 2020
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
EXAMPLE
From Jianing Song, May 21 2022: (Start)
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)
MAPLE
with(numtheory); f:=n->add( phi(d)*mobius(n/d), d in divisors(n)); [seq(f(n), n=0..120)];
MATHEMATICA
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 *)
PROG
(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
(PARI) A007431(n) = if(!n, n, my(f=factor(n)); prod(i=1, #f~, if(1==f[i, 2], f[i, 1]-2, ((f[i, 1]-1)^2)*(f[i, 1]^(f[i, 2]-2))))); \\ Antti Karttunen, Dec 15 2024, after Vladeta Jovovic's multiplicative formula
(Haskell)
a007431 0 = 0
a007431 n = sum $ map (a008683 . gcd n) [1..n]
-- Reinhard Zumkeller, Jan 06 2014
(Magma) [0] cat [&+[EulerPhi(d)*MoebiusMu(Floor(n/d)):d in Divisors(n)]:n in [1..90]]; // Marius A. Burtea, Aug 10 2019
CROSSREFS
KEYWORD
nonn,nice,mult
AUTHOR
STATUS
approved