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Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.
9

%I #14 Sep 01 2018 22:27:42

%S 1,2,9,6,25,9,49,20,243,25,121,27,169,49,225,70,289,243,361,75,441,

%T 121,529,90,1875,169,3645,147,841,225,961,252,1089,289,1225,729,1369,

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

%H Antti Karttunen, <a href="/A318649/b318649.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) * ((n^2) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

%F a(n) = n*A318512(n)*A299149(n)/A299150(n).

%o (PARI)

%o up_to = 65537;

%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 v318649_aux = DirSqrt(vector(up_to, n, (n*n)));

%o A318649(n) = numerator(v318649_aux[n]);

%Y Cf. A000290, A318512 (denominators).

%Y Cf. also A046643, A299149, A318511, A318651, A318654 (gives the positions of even terms), A318655 (the 2-adic valuation).

%K nonn

%O 1,2

%A _Antti Karttunen_, Aug 31 2018