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

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

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: A363885 A181587 A096158 * A053471 A144078 A277158

Adjacent sequences: A264779 A264780 A264781 * A264783 A264784 A264785

Keyword

nonn,easy

Author

Gevorg Hmayakyan, Nov 24 2015

Status

approved