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

1,3

Comments

This is multiplicative function with a(p^n) = |Re(I^(p^n+1) - I^(p^(n-1)+1))|.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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

Example

a(9) = |Re(I*(mobius(1)*I^9 + mobius(3)*I^3 + mobius(9)*I))| = |Re((I^10 - I^4))| = |-2| = 2.

Maple

a(n):=abs(Re(I*add(numtheory:-mobius(d)*I^(n/d), d = numtheory:-divisors(n)))).

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 *)

Prog

(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

Crossrefs

Cf. A089491, A275966.

Sequence in context: A144734 A029361 A275966 * A329767 A356018 A107502

Adjacent sequences: A284056 A284057 A284058 * A284060 A284061 A284062

Keyword

nonn,mult

Author

Gevorg Hmayakyan, Mar 19 2017

Status

approved