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A318661 Numerators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

%I A318661 #12 May 10 2025 05:30:31
%S A318661 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 A318661 29,15,31,9,33,17,35,19,37,19,39,15,41,21,43,11,95,23,47,15,123,59,51,
%U A318661 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 A318661 Numerators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.
%H A318661 Antti Karttunen, <a href="/A318661/b318661.txt">Table of n, a(n) for n = 1..65537</a>
%H A318661 Vaclav Kotesovec, <a href="/A318661/a318661.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A318661 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.
%F A318661 From _Vaclav Kotesovec_, May 10 2025: (Start)
%F A318661 Let f(s) = Product_{primes p} (1 + 1/p^(2*s) - 1/p^(2*s-1) - 1/p^s).
%F A318661 Sum_{k=1..n} A318661(k) / A318662(k) ~ n^2 * sqrt(Pi*f(2)/(24*log(n))) * (1 - ((gamma - 1)/2 + f'[2]/(2*f(2)) + 3*zeta'(2)/Pi^2) / (2*log(n))), where
%F A318661 f(2) = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.5358961538283379998085026313185459506482223745141452711510108346133288119...
%F A318661 f'(2)/f(2) = Sum_{primes p} (p^2 + 2*p - 2) * log(p) / (p^4 - p^2 - p + 1) = 0.8249574883141571786856463180997569604486048593127391054584235479395133668...
%F A318661 and gamma is the Euler-Mascheroni constant A001620. (End)
%o A318661 (PARI)
%o A318661 up_to = 1+(2^16);
%o A318661 A055653(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ From A055653
%o A318661 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 A318661 v318661_62 = DirSqrt(vector(up_to, n, A055653(n)));
%o A318661 A318661(n) = numerator(v318661_62[n]);
%o A318661 A318662(n) = denominator(v318661_62[n]);
%o A318661 A318663(n) = valuation(A318662(n),2);
%o A318661 (PARI) for(n=1, 100, print1(numerator(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 A318661 Cf. A055653, A318662 (denominators).
%K A318661 nonn,frac
%O A318661 1,3
%A A318661 _Antti Karttunen_, Sep 03 2018