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 A063659 The number of integers m in [1..n] for which gcd(m,n) is not divisible by a square greater than 1. 14
 1, 2, 3, 3, 5, 6, 7, 6, 8, 10, 11, 9, 13, 14, 15, 12, 17, 16, 19, 15, 21, 22, 23, 18, 24, 26, 24, 21, 29, 30, 31, 24, 33, 34, 35, 24, 37, 38, 39, 30, 41, 42, 43, 33, 40, 46, 47, 36, 48, 48, 51, 39, 53, 48, 55, 42, 57, 58, 59, 45, 61, 62, 56, 48, 65, 66, 67, 51, 69, 70, 71, 48 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Equals Möbius transform of A001615. - Gary W. Adamson, May 23 2008 The absolute values of the Dirichlet inverse of A007913. - R. J. Mathar, Dec 22 2010 LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 Eckford Cohen, A generalized Euler phi-function, Math. Mag. 41 (1968), 276-279; this is function phi_2(n). E. K. Haviland, An analogue of Euler's phi-function, Duke Math. J. 11 (1944), 869-872. V. L. Klee, Jr., A generalization of Euler's phi function, Amer. Math. Monthly, 55(6) (1948), 358-359; this is function Phi_2(n). Paul J. McCarthy, On a certain family of arithmetic functions, Amer. Math. Monthly 65 (1958), 586-590; this is function T_2(n). Wolfgang Schramm, The Fourier transform of functions of the greatest common divisor, Electronic Journal of Combinatorial Number Theory 8(1) (2008), #A50; see Example 5 with f(d) = mu(d)^2 (cf. the formulas by Benoit Cloitre below). FORMULA a(n) = n - A063658(n). Multiplicative with a(p) = p and a(p^e) = p^e-p^(e-2), e>1. - Vladeta Jovovic, Jul 26 2001 a(n) = Sum_{d|n} phi(d)*mu(n/d)^2, Dirichlet convolution of A000010 and A008966. - Benoit Cloitre, Sep 08 2002 a(n) = Sum_{k = 1..n} mu(gcd(n,k))^2. - Benoit Cloitre, Jun 14 2007 Dirichlet g.f.: zeta(s-1)/zeta(2s). - R. J. Mathar, Feb 27 2011 a(n) = Sum_{k=1..n} psi(gcd(k,n)) * cos(2*Pi*k/n), where psi is A001615. - Enrique Pérez Herrero, Jan 18 2013 Sum_{k=1..n} a(k) ~ 45*n^2 / Pi^4. - Vaclav Kotesovec, Jan 11 2019 [This is a special case of a general result by McCarthy (1958), which was reproved later by Cohen (1968). - Petros Hadjicostas, Jul 20 2019] EXAMPLE For n=12 we find only GCD(4,12), GCD(8,12) and GCD(12,12) divisible by 4, so a(12)=9. MAPLE A063659 := proc(n)     local a, ep, p, e;     a := 1 ;     for ep in ifactors(n) do         p := op(1, ep) ;         e := op(2, ep) ;         if e = 1 then             a := a*p ;         else             a := a*(p^e-p^(e-2)) ;         end if;     end do ;     a ; end proc: seq(A063659(n), n=1..100) ; # R. J. Mathar, Jul 04 2019 MATHEMATICA nn = 72; f[list_, i_] := list[[i]]; a =Table[If[Max[FactorInteger[n][[All, 2]]] < 2, 1, 0], {n, 1, nn}]; b =Table[EulerPhi[n], {n, 1, nn}]; Table[ DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 22 2015 *) f[p_, e_] := If[e == 1, p, p^e - p^(e-2)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 29 2020 *) PROG (PARI) a(n)=sum(k=1, n, moebius(gcd(n, k))^2) \\ Benoit Cloitre, Jun 14 2007 (PARI) for (n=1, 2000, a=1; for (m=2, n, if (issquarefree(gcd(m, n)), a++)); write("b063659.txt", n, " ", a) ) \\ Harry J. Smith, Aug 27 2009 (PARI) a(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^(f[i, 2]-2) * (f[i, 1]^2 - 1), f[i, 1])) \\ Charles R Greathouse IV, Jan 08 2018 CROSSREFS Cf. A001615. Absolute values of the Dirichlet inverse of A007913. Row 2 of A309287. Sequence in context: A097246 A277886 A337868 * A255563 A331288 A115350 Adjacent sequences:  A063656 A063657 A063658 * A063660 A063661 A063662 KEYWORD mult,nonn,easy AUTHOR Floor van Lamoen, Jul 24 2001 EXTENSIONS More terms from Vladeta Jovovic and Dean Hickerson, Jul 26 2001 Name edited by Petros Hadjicostas, Jul 21 2019 STATUS approved

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Last modified November 23 21:51 EST 2020. Contains 338603 sequences. (Running on oeis4.)