cp's OEIS Frontend

A318657 Numerators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

%I A318657 #18 Sep 23 2018 21:31:12
%S A318657 1,0,-1,0,-1,0,-1,0,-1,0,-1,0,-1,0,1,0,-1,0,-1,0,1,0,-1,0,-1,0,-1,0,
%T A318657 -1,0,-1,0,1,0,1,0,-1,0,1,0,-1,0,-1,0,1,0,-1,0,-1,0,1,0,-1,0,1,0,1,0,
%U A318657 -1,0,-1,0,1,0,1,0,-1,0,1,0,-1,0,-1,0,1,0,1,0,-1,0,-5,0,-1,0,1,0,1,0,-1,0,1,0,1,0,1,0,-1,0,1,0,-1,0,-1,0,-1
%N A318657 Numerators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).
%C A318657 Because the corresponding denominator sequence A318658 is equal to A046644 on all odd n, and this sequence as well as A087003 is zero on all even n, it means that also the Dirichlet convolution of a(n)/A046644(n) with itself will yield A087003. Because both A046644 and A087003 are multiplicative, this sequence is also. - _Antti Karttunen_, Sep 01 2018
%H A318657 Antti Karttunen, <a href="/A318657/b318657.txt">Table of n, a(n) for n = 1..65537</a>
%F A318657 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A087003(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318657 a(2n) = 0, a(2n-1) = A257098(2n-1), thus multiplicative with a(2^e) = 0, a(p^e) = A257098(p^e) for odd primes p.  - _Antti Karttunen_, Sep 01 2018
%o A318657 (PARI)
%o A318657 up_to = 65537;
%o A318657 A087003(n) = ((n%2)*moebius(n)); \\ I.e. a(n) = A000035(n)*A008683(n).
%o A318657 DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&d<n, u[d]*u[n/d], 0)))/2); u};
%o A318657 v318657_18 = DirSqrt(vector(up_to, n, A087003(n)));
%o A318657 A318657(n) = numerator(v318657_18[n]);
%Y A318657 Cf. A046644 or A318658 (denominators).
%Y A318657 Cf. also A087003, A257098, A318659.
%K A318657 sign,frac,mult
%O A318657 1,81
%A A318657 _Antti Karttunen_, Aug 31 2018