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A340144 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).

%I A340144 #10 May 09 2025 03:34:45
%S A340144 1,1,1,7,1,3,1,25,11,7,1,19,1,11,3,363,1,31,1,43,5,19,1,63,19,23,61,3,
%T A340144 1,19,1,1335,9,31,11,189,1,35,11,139,1,29,1,115,7,43,1,867,27,127,15,
%U A340144 11,1,163,19,279,17,55,1,93,1,59,51,9923,5,-15,1,187,21,3,1,615,1,71,55,19,29,59,1,1875,1363,79,1,203
%N A340144 Numerators of the sequence whose Dirichlet convolution with itself yields sequence A247074(x) = phi(x)/(Product_{primes p dividing x} gcd(p-1, x-1)).
%H A340144 Antti Karttunen, <a href="/A340144/b340144.txt">Table of n, a(n) for n = 1..8191</a>
%H A340144 Antti Karttunen, <a href="/A340144/a340144.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%e A340144 For n = 561 = 3*11*17, its divisors d are: 1, 3, 11, 17, 33, 51, 187, 561.
%e A340144 For this sequence, the corresponding terms a(d) are: 1, 1, 1, 1, 9, 15, 79, -99.
%e A340144 For A046644, the corresponding terms are:            1, 2, 2, 2, 4,  4,  4,   8.
%e A340144 Convolving these ratios as Sum_{d|561} r(d)*r(n/d) = 2*((1/1)*(-99/8) + (1/2)*(79/4) + (1/2)*(15/4) + (1/2)*(9/4)) yields 1 as expected, because 561 is Carmichael number (A002997) and A247074 obtains value 1 on all of them.
%o A340144 (PARI)
%o A340144 up_to = 65537;
%o A340144 A247074(n) = { my(f=factor(n)); eulerphi(f)/prod(i=1, #f~, gcd(f[i, 1]-1, n-1)); }; \\ From A247074
%o A340144 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 A340144 v340144rat = DirSqrt(vector(up_to, n, A247074(n)));
%o A340144 A340144(n) = numerator(v340144rat[n]);
%Y A340144 Cf. A046644 (denominators).
%Y A340144 Cf. A247074.
%Y A340144 Cf. also A340141, A340145, A340146.
%K A340144 sign,frac
%O A340144 1,4
%A A340144 _Antti Karttunen_, Dec 29 2020