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A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.

%I A318649 #21 May 09 2025 08:59:54
%S A318649 1,2,9,6,25,9,49,20,243,25,121,27,169,49,225,70,289,243,361,75,441,
%T A318649 121,529,90,1875,169,3645,147,841,225,961,252,1089,289,1225,729,1369,
%U A318649 361,1521,250,1681,441,1849,363,6075,529,2209,315,7203,1875,2601,507,2809,3645,3025,490,3249,841,3481,675,3721,961,11907,924,4225,1089
%N A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.
%H A318649 Antti Karttunen, <a href="/A318649/b318649.txt">Table of n, a(n) for n = 1..65537</a>
%H A318649 Vaclav Kotesovec, <a href="/A318649/a318649.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A318649 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * ((n^2) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%F A318649 a(n) = n*A318512(n)*A299149(n)/A299150(n).
%F A318649 Sum_{k=1..n} A318649(k) / A318512(k) ~ n^3/(3*sqrt(Pi*log(n))) * (1 + (1 - 3*gamma/2) / (6*log(n))), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 09 2025
%o A318649 (PARI)
%o A318649 up_to = 65537;
%o A318649 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 A318649 v318649_aux = DirSqrt(vector(up_to, n, (n*n)));
%o A318649 A318649(n) = numerator(v318649_aux[n]);
%o A318649 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p^2*X)^(1/2))[n]), ", ")) \\ _Vaclav Kotesovec_, May 09 2025
%Y A318649 Cf. A000290, A318512 (denominators).
%Y A318649 Cf. also A046643, A299149, A318511, A318651, A318654 (gives the positions of even terms), A318655 (the 2-adic valuation).
%K A318649 nonn,frac
%O A318649 1,2
%A A318649 _Antti Karttunen_, Aug 31 2018