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 A264782 a(n) = Sum_{d|n} möbius(d)^(n/d). 2
 1, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 2, 0, 8, 0, 2, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 FORMULA a(n) = Sum_{d|n} möbius(d)^(n/d). For odd n, a(n)=0. For n = 2 * p1^k1 * p2^k2 * ... * pr^kr, a(n) = 2^r. For n = 2^m * p1^k1 * p2^k2 * ... * pr^kr, a(n) = 2^(r+1) if m > 1. a(2n) = A034444(n) for n > 1. From Gevorg Hmayakyan, Dec 31 2016: (Start) If b(n) = Sum_{d|n} möbius(d)^d, then b(n) = (A209229(n)+1)*((-1)^n + 1)/2*a(2*n)/2, for n > 1. Dirichlet g.f.: -1 + 2^(-s) + (2^(-s) Zeta[s]^2)/Zeta[2s]. (End) Sum_{k=1..n} a(k) ~ 3*n / Pi^2 * (log(n) - 1 + 2*gamma - log(2) - 12*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019 G.f.: Sum_{k>=1} mu(k)*x^k/(1 - mu(k)*x^k). - Ilya Gutkovskiy, May 23 2019 EXAMPLE a(1) = 1, a(p) = mu(1)^p + mu(p)^1 = 0. a(p1*p2) = mu(1)^p1*p2 + mu(p1)^p2 + mu(p2)^p1 + mu(p1*p2) = 1+(-1)+(-1)+1 = 0. a(2*p) = mu(1)^2*p + mu(2)^p + mu(p)^2 + mu(2*p) = 1+(-1)+1+1 = 2. MATHEMATICA Table[Sum[MoebiusMu[d]^(n/d), {d, Divisors@ n}], {n, 87}] (* Michael De Vlieger, Nov 25 2015 *) PROG (PARI) a(n) = sumdiv(n, d, moebius(n/d)^d); (Haskell) a264782 n = sum \$ zipWith (^) (map a008683 divs) (reverse divs)             where divs = a027750_row n -- Reinhard Zumkeller, Dec 19 2015 (Perl) use ntheory ":all"; sub a264782 { my \$n=shift; divisor_sum(\$n, sub { moebius(\$_[0]) ** (\$n/\$_[0]) }); } # Dana Jacobsen, Dec 29 2015 CROSSREFS Cf. A008683, A027750, A034444. Sequence in context: A045843 A181587 A096158 * A053471 A144078 A277158 Adjacent sequences:  A264779 A264780 A264781 * A264783 A264784 A264785 KEYWORD nonn,easy AUTHOR Gevorg Hmayakyan, Nov 24 2015 STATUS approved

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Last modified August 24 03:51 EDT 2019. Contains 326260 sequences. (Running on oeis4.)