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
Examples
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.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Programs
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Haskell
a264782 n = sum $ zipWith (^) (map a008683 divs) (reverse divs) where divs = a027750_row n -- Reinhard Zumkeller, Dec 19 2015 -
Mathematica
Table[Sum[MoebiusMu[d]^(n/d), {d, Divisors@ n}], {n, 87}] (* Michael De Vlieger, Nov 25 2015 *) -
PARI
a(n) = sumdiv(n, d, moebius(n/d)^d);
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Perl
use ntheory ":all"; sub a264782 { my $n=shift; divisor_sum($n, sub { moebius($[0]) ** ($n/$[0]) }); } # Dana Jacobsen, Dec 29 2015
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(2*s) [corrected by Michael Shamos, Jul 18 2025]. (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