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A318662 Denominators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

%I A318662 #10 May 10 2025 05:26:31
%S A318662 1,1,2,1,2,2,2,2,8,2,2,2,2,2,4,2,2,8,2,2,4,2,2,4,8,2,16,2,2,4,2,2,4,2,
%T A318662 4,8,2,2,4,4,2,4,2,2,16,2,2,4,8,8,4,2,2,16,4,4,4,2,2,4,2,2,16,8,4,4,2,
%U A318662 2,4,4,2,16,2,2,16,2,4,4,2,4,128,2,2,4,4,2,4,4,2,16,4,2,4,2,4,4,2,8,16,8,2,4,2,4,8
%N A318662 Denominators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.
%C A318662 The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".
%H A318662 Antti Karttunen, <a href="/A318662/b318662.txt">Table of n, a(n) for n = 1..16384</a>
%F A318662 a(n) = denominator 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 A318662 (PARI)
%o A318662 up_to = 1+(2^16);
%o A318662 A055653(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ From A055653
%o A318662 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 A318662 v318661_62 = DirSqrt(vector(up_to, n, A055653(n)));
%o A318662 A318661(n) = numerator(v318661_62[n]);
%o A318662 A318662(n) = denominator(v318661_62[n]);
%o A318662 A318663(n) = valuation(A318662(n),2);
%o A318662 (PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, ((1 + X^2 - p*X^2 - X)/((1-X)*(1-p*X)))^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 10 2025
%Y A318662 Cf. A055653, A318661 (numerators), A318663.
%Y A318662 Cf. also A046644, A299150, A317932, A317934, A318314.
%K A318662 nonn,frac
%O A318662 1,3
%A A318662 _Antti Karttunen_, Sep 03 2018