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A317847 Numerators of sequence whose Dirichlet convolution with itself yields A303757, the ordinal transform of function a(1) = 0; a(n) = phi(n) for n > 1, where phi is Euler's totient function (A000010).

%I A317847 #13 May 09 2025 03:35:34
%S A317847 1,1,1,7,1,5,1,9,7,5,1,15,1,5,1,43,1,15,1,7,3,3,1,5,3,5,9,15,1,9,1,87,
%T A317847 3,5,1,1,1,5,3,13,1,11,1,11,15,3,1,187,7,19,1,15,1,5,3,21,3,3,1,-1,1,
%U A317847 3,11,387,1,9,1,7,1,13,1,119,1,7,19,23,3,19,1,139,-21,7,1,21,1,5,1,39,1,67,3,3,5,3,5,451,1,15,19,69,1,13,1,-27,7
%N A317847 Numerators of sequence whose Dirichlet convolution with itself yields A303757, the ordinal transform of function a(1) = 0; a(n) = phi(n) for n > 1, where phi is Euler's totient function (A000010).
%H A317847 Antti Karttunen, <a href="/A317847/b317847.txt">Table of n, a(n) for n = 1..65537</a>
%F A317847 a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A303757(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
%t A317847 A303757[n_] := If[n == 2, 1, Count[EulerPhi[Range[n]] - EulerPhi[n], 0]];
%t A317847 f[n_] := f[n] = If[n == 1, 1, (1/2)(A303757[n] -
%t A317847      Sum[If[1<d<n, f[d] f[n/d], 0], {d, Divisors[n]}])];
%t A317847 a[n_] := Numerator[f[n]];
%t A317847 Array[a, 105] (* _Jean-François Alcover_, Dec 20 2021 *)
%o A317847 (PARI)
%o A317847 up_to = 65537;
%o A317847 ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
%o A317847 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 A317847 v303757 = ordinal_transform(vector(up_to,n,if(1==n,0,eulerphi(n))));
%o A317847 v317847 = DirSqrt(vector(up_to, n, v303757[n]));
%o A317847 A317847(n) = numerator(v317847[n]);
%Y A317847 Cf. A000010, A303757, A046644 (denominators).
%Y A317847 Cf. also A317830, A317833, A317834, A317937.
%K A317847 sign,frac
%O A317847 1,4
%A A317847 _Antti Karttunen_, Aug 14 2018