A120630 Dirichlet inverse of A002654.

1, -1, 0, 0, -2, 0, 0, 0, -1, 2, 0, 0, -2, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, -1, -1, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0

Offset: 1

Author

Gerard P. Michon, Jun 25 2006

Keywords

• mult
• easy
• sign

Examples

			a(65)=4 because 65 is 5 times 13 and both of those primes are congruent to 1 modulo 4. Doubling an odd index yields the opposite of the value (e.g., a(130)=-4) because a(2)=-1. Doubling an even index yields zero.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A002654, A006752, A023900, A046692, A053822, A053825, A053826, A101035.

Programs

Maple

A120630 := proc(n)
    local a,pp;
    if n = 1 then
        1;
    else
        a := 1 ;
        for pp in ifactors(n)[2] do
            if op(2,pp) > 2 then
                a := 0;
            elif op(1,pp) = 2 then
                if op(2,pp) = 1 then
                    a := -a ;
                else
                    a := 0 ;
                end if;
            elif modp(op(1,pp),4) = 3 then
                if op(2,pp) = 1 then
                    a := 0 ;
                else
                    a := -a ;
                end if;
            else
                if op(2,pp) = 1 then
                    a := -2*a ;
                else
                    ;
                end if;
            end if;
        end do:
        a;
    end if;
end proc: # R. J. Mathar, Sep 15 2015

Mathematica

A120630[n_] := Module[{a, pp}, If[n == 1, 1, a = 1; Do[Which[pp[[2]] > 2, a = 0, pp[[1]] == 2, If[pp[[2]] == 1, a = -a, a = 0], Mod[pp[[1]], 4] == 3, If[pp[[2]] == 1, a = 0, a = -a], True, If[pp[[2]] == 1, a = -2*a]], {pp, FactorInteger[n]}]; a]]; Array[A120630, 120] (* Jean-François Alcover, Apr 24 2017, after R. J. Mathar *)

PARI

seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sumdiv( n, d, (d%4==1) - (d%4==3))))} \\ Andrew Howroyd, Aug 05 2018

Formula

Multiplicative function with a(p^e)=0 if e>2. a(2)=-1, a(4)=0. If p is a prime congruent to 3 modulo 4, then a(p)=0 and a(p^2)=-1. If p is a prime congruent to 1 modulo 4, then a(p)=-2 and a(p^2)=1.

Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 3/(2*Pi*G) = 0.521269..., and G is Catalan's constant (A006752). - Amiram Eldar, Jan 22 2024