A317936 - OEIS

%I #7 Aug 23 2018 21:01:54

%S 1,1,1,7,1,-1,1,17,7,-1,1,-5,1,-1,-1,139,1,-5,1,-5,-1,-1,1,-5,7,-1,17,

%T -5,1,3,1,263,-1,-1,-1,-31,1,-1,-1,-5,1,3,1,-5,-5,-1,1,19,7,-5,-1,-5,

%U 1,-5,-1,-5,-1,-1,1,9,1,-1,-5,995,-1,3,1,-5,-1,3,1,-53,1,-1,-5,-5,-1,3,1,19,139,-1,1,9,-1,-1,-1,-5,1,9

%N Numerators of sequence whose Dirichlet convolution with itself yields A100995 + A063524, that is, the characteristic function of A000961 (prime powers).

%H Antti Karttunen, <a href="/A317936/b317936.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A100995(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

%o (PARI)

%o up_to = 65537;

%o 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}; \\ From A317937.

%o v317936aux = DirSqrt(vector(up_to, n, if(1==n,1,isprimepower(n))));

%o A317936(n) = numerator(v317936aux[n]);

%o for(n=1,up_to,write("b317936.txt", n, " ", A317936(n)));

%Y Cf. A000961, A100995, A046644 (denominators).

%Y Cf. also A317939.

%K sign,frac

%O 1,4

%A _Antti Karttunen_, Aug 14 2018