%I A129667
%S A129667 1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,1,1,0,1,1,1,1,1,1,1,1,
%T A129667 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,
%U A129667 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
%V 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,-1,
               -1,-1,1,1,1,1,
%W A129667 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,
%X A129667 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 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#partitions">
               Partition Function</a> and Pentagonal Numbers.
%H A129667 G. P. Michon, <a href="http://www.numericana.com/answer/numbers.htm#multiplicative">
               Multiplicative Functions</a>.
%F A129667 Multiplicative function for which a(p^e) either vanishes or is equal 
               to (-1)^n, for any prime p, if e is either n(3n-1)/2 or n(3n+1)/2 
               (these integers are the pentagonal numbers of the first and second 
               kind, A000326 and A005449).
%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.
%Y A129667 Cf. A000041, A000326, A000688, A005449, A023900, A101035.
%Y A129667 Sequence in context: A119981 A115789 A053864 this_sequence A071374 A077010 
               A166280
%Y A129667 Adjacent sequences: A129664 A129665 A129666 this_sequence A129668 A129669 
               A129670
%K A129667 mult,easy,sign
%O A129667 1,1
%A A129667 Gerard P. Michon (g.michon(AT)att.net), Apr 28 2007, May 01 2007