A129667 Dirichlet inverse of the Abelian group count (A000688).

1, -1, -1, -1, -1, 1, -1, 0, -1, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, 1, 1, -1, 0, -1, 1, 0, 1, -1, -1, -1, 1, 1, 1, 1, 1, -1, 1, 1, 0, -1, -1, -1, 1, 1, 1, -1, 0, -1, 1, 1, 1, -1, 0, 1, 0, 1, 1, -1, -1, -1, 1, 1, 0, 1, -1, -1, 1, 1, -1, -1, 0, -1, 1, 1, 1, 1, -1, -1, 0, 0, 1, -1, -1, 1, 1, 1, 0, -1, -1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, -1, -1, -1, 0, -1, 1, -1

Offset: 1

Author

Gerard P. Michon, Apr 28 2007, May 01 2007

Keywords

• mult
• easy
• sign

Comments

The simple formula which gives the value of this multiplicative function for the power of any prime can be derived from Euler's celebrated "Pentagonal Number Theorem" (applied to the generating function of the partition function A000041 on which A000688 is based).

Examples

			a(8) and a(27) are zero because the sequence vanishes for the cubes of primes. Not so with fifth powers of primes (since 5 is a pentagonal number) so a(32) is nonzero.
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..1000
• Gérard P. Michon, Multiplicative Functions.
• Gérard P. Michon, Partition Function and Pentagonal Numbers.

Crossrefs

Cf. A000041, A000326, A000688, A005449, A023900, A101035.

Programs

Maple

A000326inv := proc(n)
    local x,a ;
    for x from 0 do
        a := x*(3*x-1)/2 ;
        if a > n then
            return -1 ;
        elif a = n then
            return x;
        end if;
    end do:
end proc:
A005449inv := proc(n)
    local x,a ;
    for x from 0 do
        a := x*(3*x+1)/2 ;
        if a > n then
            return -1 ;
        elif a = n then
            return x;
        end if;
    end do:
end proc:
A129667 := proc(n)
    local a,e1,e2 ;
    a := 1 ;
    for pe in ifactors(n)[2] do
        e1 := A000326inv(op(2,pe)) ;
        e2 := A005449inv(op(2,pe)) ;
        if e1 >= 0 then
            a := a*(-1)^e1 ;
        elif e2 >= 0 then
            a := a*(-1)^e2 ;
        else
            a := 0 ;
        end if;
    end do:
    a;
end proc: # R. J. Mathar, Nov 24 2017

Mathematica

a[n_] := a[n] = If[n == 1, 1, -Sum[FiniteAbelianGroupCount[n/d] a[d], {d, Most @ Divisors[n]}]];
Array[a, 100] (* Jean-François Alcover, Feb 16 2020 *)

Formula

Multiplicative function for which a(p^e) either vanishes or is equal to (-1)^m, for any prime p, if e is either m(3m-1)/2 or m(3m+1)/2 (these integers are the pentagonal numbers of the first and second kind, A000326 and A005449).

Dirichlet g.f.: 1 / Product_{k>=1} zeta(k*s). - Ilya Gutkovskiy, Nov 06 2020

Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Product_{p prime} ((1-1/p) * (1 + Sum_{m>=1} (1/p^(m*(3*m-1)/2) + 1/p^(m*(3*m+1)/2)))) = 0.85358290653064143678... . - Amiram Eldar, Feb 17 2024