This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A318653 #18 May 08 2025 13:45:49
%S A318653 1,1,3,1,5,3,7,1,3,5,11,3,13,7,15,3,17,3,19,5,21,11,23,3,-5,13,15,7,
%T A318653 29,15,31,3,33,17,35,3,37,19,39,5,41,21,43,11,15,23,47,9,-21,-5,51,13,
%U A318653 53,15,55,7,57,29,59,15,61,31,21,5,65,33,67,17,69,35,71,3,73,37,-15,19,77,39,79,15,3,41,83,21,85,43,87,11,89,15
%N A318653 Numerators of the sequence whose Dirichlet convolution with itself yields A007947, the squarefree kernel of n.
%C A318653 No zeros among the first 2^20 terms.
%H A318653 Antti Karttunen, <a href="/A318653/b318653.txt">Table of n, a(n) for n = 1..65537</a>
%H A318653 Vaclav Kotesovec, <a href="/A318653/a318653.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A318653 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A007947(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318653 From _Vaclav Kotesovec_, May 08 2025: (Start)
%F A318653 Let f(s) = Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(2*s-2) - 1/p^s).
%F A318653 Sum_{k=1..n} A318653(k)/A299150(k) ~ n^2 * sqrt(Pi*f(2)/(24*log(n))) * (1 - (gamma - 1 + f'(2)/f(2) + 6*zeta'(2)/Pi^2) / (4*log(n))), where
%F A318653 f(2) = A065464 = Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.4282495056770944402187657075818235461212985133559361440319...
%F A318653 f'(2) = f(2) * Sum_{p prime} (3*p-2)*log(p) / ((p-1)*(p^2+p-1)) = f(2) * 1.469536740824614833203393993450164364663334798759143895712...
%F A318653 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A318653 rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[1] = 1; f[n_] := f[n] = (rad[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; a[n_] := Numerator [f[n]]; Array[a, 100] (* _Amiram Eldar_, Dec 07 2020 *)
%o A318653 (PARI)
%o A318653 up_to = 65537;
%o A318653 A007947(n) = factorback(factorint(n)[, 1]);
%o A318653 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 A318653 v318653_aux = DirSqrt(vector(up_to, n, A007947(n)));
%o A318653 A318653(n) = numerator(v318653_aux[n]);
%o A318653 for(n=1, 100, print1(numerator(direuler(p=2, n, ((1 + p*X - X)/(1 - X))^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 08 2025
%Y A318653 Cf. A007947, A299150 (denominators).
%Y A318653 Cf. also A317935, A318511, A318512, A318649.
%K A318653 sign,frac,mult
%O A318653 1,3
%A A318653 _Antti Karttunen_, Aug 31 2018