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

%I A129667 #17 Feb 17 2024 04:03:55
%S A129667 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,
%T A129667 -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,
%U A129667 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
%N A129667 Dirichlet inverse of the Abelian group count (A000688).
%C A129667 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).
%H A129667 R. J. Mathar, <a href="/A129667/b129667.txt">Table of n, a(n) for n = 1..1000</a>
%H A129667 Gérard P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">Multiplicative Functions</a>.
%H A129667 Gérard P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#partitions">Partition Function</a> and Pentagonal Numbers.
%F A129667 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).
%F A129667 Dirichlet g.f.: 1 / Product_{k>=1} zeta(k*s). - _Ilya Gutkovskiy_, Nov 06 2020
%F A129667 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
%e A129667 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.
%p A129667 A000326inv := proc(n)
%p A129667     local x,a ;
%p A129667     for x from 0 do
%p A129667         a := x*(3*x-1)/2 ;
%p A129667         if a > n then
%p A129667             return -1 ;
%p A129667         elif a = n then
%p A129667             return x;
%p A129667         end if;
%p A129667     end do:
%p A129667 end proc:
%p A129667 A005449inv := proc(n)
%p A129667     local x,a ;
%p A129667     for x from 0 do
%p A129667         a := x*(3*x+1)/2 ;
%p A129667         if a > n then
%p A129667             return -1 ;
%p A129667         elif a = n then
%p A129667             return x;
%p A129667         end if;
%p A129667     end do:
%p A129667 end proc:
%p A129667 A129667 := proc(n)
%p A129667     local a,e1,e2 ;
%p A129667     a := 1 ;
%p A129667     for pe in ifactors(n)[2] do
%p A129667         e1 := A000326inv(op(2,pe)) ;
%p A129667         e2 := A005449inv(op(2,pe)) ;
%p A129667         if e1 >= 0 then
%p A129667             a := a*(-1)^e1 ;
%p A129667         elif e2 >= 0 then
%p A129667             a := a*(-1)^e2 ;
%p A129667         else
%p A129667             a := 0 ;
%p A129667         end if;
%p A129667     end do:
%p A129667     a;
%p A129667 end proc: # _R. J. Mathar_, Nov 24 2017
%t A129667 a[n_] := a[n] = If[n == 1, 1, -Sum[FiniteAbelianGroupCount[n/d] a[d], {d, Most @ Divisors[n]}]];
%t A129667 Array[a, 100] (* _Jean-François Alcover_, Feb 16 2020 *)
%Y A129667 Cf. A000041, A000326, A000688, A005449, A023900, A101035.
%K A129667 mult,easy,sign
%O A129667 1,1
%A A129667 _Gerard P. Michon_, Apr 28 2007, May 01 2007