A318661 - OEIS

%I #5 Sep 03 2018 23:01:25

%S 1,1,3,1,5,3,7,3,19,5,11,3,13,7,15,5,17,19,19,5,21,11,23,9,59,13,95,7,

%T 29,15,31,9,33,17,35,19,37,19,39,15,41,21,43,11,95,23,47,15,123,59,51,

%U 13,53,95,55,21,57,29,59,15,61,31,133,67,65,33,67,17,69,35,71,57,73,37,177,19,77,39,79,25,2019,41,83,21,85,43,87,33,89

%N Numerators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

%H Antti Karttunen, <a href="/A318661/b318661.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) * (A055653(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

%o (PARI)

%o up_to = 1+(2^16);

%o A055653(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ From A055653

%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};

%o v318661_62 = DirSqrt(vector(up_to, n, A055653(n)));

%o A318661(n) = numerator(v318661_62[n]);

%o A318662(n) = denominator(v318661_62[n]);

%o A318663(n) = valuation(A318662(n),2);

%Y Cf. A055653, A318662 (denominators).

%K nonn

%O 1,3

%A _Antti Karttunen_, Sep 03 2018