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A264782 a(n) = Sum_{d|n} möbius(d)^(n/d).

%I A264782 #47 Jul 19 2025 03:14:06
%S A264782 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,
%T A264782 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,
%U A264782 0,4,0,4,0,2,0,4,0,4,0,4,0,2,0,8,0,2,0
%N A264782 a(n) = Sum_{d|n} möbius(d)^(n/d).
%H A264782 Reinhard Zumkeller, <a href="/A264782/b264782.txt">Table of n, a(n) for n = 1..10000</a>
%F A264782 a(n) = Sum_{d|n} möbius(d)^(n/d).
%F A264782 For odd n, a(n)=0.
%F A264782 For n = 2 * p1^k1 * p2^k2 * ... * pr^kr, a(n) = 2^r.
%F A264782 For n = 2^m * p1^k1 * p2^k2 * ... * pr^kr, a(n) = 2^(r+1) if m > 1.
%F A264782 a(2n) = A034444(n) for n > 1.
%F A264782 From _Gevorg Hmayakyan_, Dec 31 2016: (Start)
%F A264782 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.
%F A264782 Dirichlet g.f.: 1 - 2^(-s) + 2^(-s)*zeta(s)^2/zeta(2*s) [corrected by _Michael Shamos_, Jul 18 2025]. (End)
%F A264782 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
%F A264782 G.f.: Sum_{k>=1} mu(k)*x^k/(1 - mu(k)*x^k). - _Ilya Gutkovskiy_, May 23 2019
%e A264782 a(1) = 1, a(p) = mu(1)^p + mu(p)^1 = 0.
%e A264782 a(p1*p2) = mu(1)^p1*p2 + mu(p1)^p2 + mu(p2)^p1 + mu(p1*p2) = 1+(-1)+(-1)+1 = 0.
%e A264782 a(2*p) = mu(1)^2*p + mu(2)^p + mu(p)^2 + mu(2*p) = 1+(-1)+1+1 = 2.
%t A264782 Table[Sum[MoebiusMu[d]^(n/d), {d, Divisors@ n}], {n, 87}] (* _Michael De Vlieger_, Nov 25 2015 *)
%o A264782 (PARI) a(n) = sumdiv(n, d, moebius(n/d)^d);
%o A264782 (Haskell)
%o A264782 a264782 n = sum $ zipWith (^) (map a008683 divs) (reverse divs)
%o A264782             where divs = a027750_row n
%o A264782 -- _Reinhard Zumkeller_, Dec 19 2015
%o A264782 (Perl) use ntheory ":all"; sub a264782 { my $n=shift; divisor_sum($n, sub { moebius($_[0]) ** ($n/$_[0]) }); } # _Dana Jacobsen_, Dec 29 2015
%Y A264782 Cf. A008683, A027750, A034444.
%K A264782 nonn,easy
%O A264782 1,4
%A A264782 _Gevorg Hmayakyan_, Nov 24 2015