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 A129667 Dirichlet inverse of the Abelian group count (A000688). 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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). LINKS R. J. Mathar, Table of n, a(n) for n = 1..1000 G. P. Michon, Partition Function and Pentagonal Numbers. G. P. Michon, Multiplicative Functions. 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). EXAMPLE 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. 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 CROSSREFS Cf. A000041, A000326, A000688, A005449, A023900, A101035. Sequence in context: A053864 A189021 A307420 * A071374 A071025 A077010 Adjacent sequences:  A129664 A129665 A129666 * A129668 A129669 A129670 KEYWORD mult,easy,sign AUTHOR Gerard P. Michon, Apr 28 2007, May 01 2007 STATUS approved

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Last modified October 23 20:17 EDT 2019. Contains 328373 sequences. (Running on oeis4.)