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%I #10 Sep 13 2018 06:22:07
%S 1,3,7,35,21,21,43,239,195,63,111,245,157,129,147,6851,273,585,343,
%T 735,301,333,507,1673,1643,471,3011,1505,813,441,931,50141,777,819,
%U 903,6825,1333,1029,1099,5019,1641,903,1807,3885,4095,1521,2163,47957,6555,4929,1911,5495,2757,9033,2331,10277,2401,2439,3423
%N Numerators of the sequence whose Dirichlet convolution with itself yields A057660(n) = Sum_{k=1..n} n/gcd(n,k).
%C Because A057660 contains only odd values, A046644 gives the sequence of denominators. Because both of those sequences are multiplicative, this is also.
%H Antti Karttunen, <a href="/A318444/b318444.txt">Table of n, a(n) for n = 1..16384</a>
%F a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057660(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%t a57660[n_] := DivisorSigma[2, n^2]/DivisorSigma[1, n^2];
%t f[1] = 1; f[n_] := f[n] = 1/2 (a57660[n] - Sum[f[d]*f[n/d], {d, Divisors[ n][[2 ;; -2]]}]);
%t Table[f[n] // Numerator, {n, 1, 60}] (* _Jean-François Alcover_, Sep 13 2018 *)
%o (PARI)
%o up_to = 16384;
%o A057660(n) = sumdivmult(n, d, eulerphi(d)*d); \\ From A057660
%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}; \\ From A317937.
%o v318444aux = DirSqrt(vector(up_to, n, A057660(n)));
%o A318444(n) = numerator(v318444aux[n]);
%Y Cf. A057660, A046644 (denominators).
%Y Cf. also A318443.
%K nonn,frac,mult
%O 1,2
%A _Antti Karttunen_ and _Andrew Howroyd_, Aug 29 2018
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