A284059 The absolute values of A275966.
1, 1, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 4, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 2, 0, 0, 2, 4, 0, 0, 4, 2, 0, 4, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0
Offset: 1
Examples
a(9) = |Re(I*(mobius(1)*I^9 + mobius(3)*I^3 + mobius(9)*I))| = |Re((I^10 - I^4))| = |-2| = 2.
Links
- Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Programs
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Maple
a(n):=abs(Re(I*add(numtheory:-mobius(d)*I^(n/d), d = numtheory:-divisors(n)))).
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Mathematica
Table[Abs@ Re[I* Sum[MoebiusMu[d] * I^(n/d), {d, Divisors[n]}]], {n, 87}] (* Indranil Ghosh, Mar 19 2017 *) f[p_, e_] := If[Mod[p, 4] == 1, 0, 2]; f[2, e_] := If[e == 1, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 27 2024 *) -
PARI
a(n)=my(f=factor(n)); prod(i=1,#f~, if(f[i,1]==2, if(f[i,2]==1,1,0), if(f[i,1]%4==3, 2, 0))) \\ Charles R Greathouse IV, Mar 22 2017
Formula
a(n) = |Re(I*Sum_{d|n}(mobius(d)*I^(n/d)))|.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/Pi = 0.954929... (A089491). - Amiram Eldar, Jan 27 2024
Comments