|
|
A264782
|
|
a(n) = Sum_{d|n} möbius(d)^(n/d).
|
|
3
|
|
|
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
|
|
|
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.
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
|
|
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
(Perl) use ntheory ":all"; sub a264782 { my $n=shift; divisor_sum($n, sub { moebius($_[0]) ** ($n/$_[0]) }); } # Dana Jacobsen, Dec 29 2015
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|