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%I #9 Feb 27 2024 03:55:58
%S 1,1,3,1,7,3,3,7,1,7,9,7,14,3,15,1,31,1,39,5,7,3,9,15,7,7,39,7,7,5,9,
%T 31,21,31,15,7,91,39,5,31,15,7,24,7,5,3,9,31,8,7,21,10,49,39,15,15,91,
%U 7,45,31,28,9,91,1,31,7,9,7,21,5,6,5,65,91,3,91,21
%N Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.
%H Amiram Eldar, <a href="/A370689/b370689.txt">Table of n, a(n) for n = 1..10000</a>
%H Jean-Marie De Koninck and Florian Luca, <a href="https://eudml.org/doc/284003">On the composition of the Euler function and the sum of divisors function</a>, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.
%F Let f(n) = a(n)/A370690(n) = A062402(n)/A062401(n).
%F Formulas from De Koninck and Luca (2007):
%F lim sup_{n->oo} f(n)/log_2(n)^2 = exp(2*gamma) (A091724).
%F lim inf_{n->oo} f(n)/log_2(n)^2 = delta exists, and exp(-gamma)/40 <= delta <= 2*exp(-gamma).
%F Sum_{k=1..n} f(k) = c * exp(2*gamma) * log_3(n)^2 * n + O(n * log_3(n)^(3/2)), where c = Product_{p prime} (1 - 3/(p*(p + 1)) + 1/(p^2*(p + 1)) + ((p-1)^3/p^2)*Sum_{k>=3} 1/(p^k-1)) = 0.45782563109026414241... .
%e Fractions begin with: 1, 1/2, 3/2, 1/2, 7/2, 3/4, 3, 7/8, 1, 7/6, 9/2, 7/12, ...
%t Table[DivisorSigma[1, EulerPhi[n]]/EulerPhi[DivisorSigma[1, n]], {n, 1, 100}] // Numerator
%o (PARI) a(n) = {my(f = factor(n)); numerator(sigma(eulerphi(f)) / eulerphi(sigma(f)));}
%Y Cf. A000010, A000203, A033632, A062401, A062402, A065395, A066930, A289336, A073858 (positions of 1's), A289412, A370690 (denominators).
%Y Cf. A080130, A091724.
%K nonn,easy,frac
%O 1,3
%A _Amiram Eldar_, Feb 27 2024
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