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A007431 a(n) = Sum_{d|n} phi(d)*mu(n/d).
(Formerly M2197)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Also Möbius 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

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

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^e-2*p^(e-1)+p^(e-2) for e>1. - Vladeta Jovovic, Jan 25 2002

Dirichlet g.f.: zeta(x-1)/zeta^2(x).

n>0, a(n) = Sum_{k=1..n} mu(gcd(n,k)). - 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

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 *)

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

(Haskell)

a007431 0 = 0

a007431 n = sum $ map (a008683 . gcd n) [1..n]

-- Reinhard Zumkeller, Jan 06 2014

CROSSREFS

Cf. A007432.

Cf. A008683.

Sequence in context: A236146 A196111 A261628 * A215447 A159980 A098496

Adjacent sequences:  A007428 A007429 A007430 * A007432 A007433 A007434

KEYWORD

nonn,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 22 22:17 EDT 2017. Contains 283901 sequences.