A370689 Numerator of sigma(phi(n))/phi(sigma(n)), where sigma is the sum of the divisors function and phi is the Euler totient function.

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, 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, 7, 45, 31, 28, 9, 91, 1, 31, 7, 9, 7, 21, 5, 6, 5, 65, 91, 3, 91, 21

Offset: 1

Author

Amiram Eldar, Feb 27 2024

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with: 1, 1/2, 3/2, 1/2, 7/2, 3/4, 3, 7/8, 1, 7/6, 9/2, 7/12, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Jean-Marie De Koninck and Florian Luca, On the composition of the Euler function and the sum of divisors function, Colloquium Mathematicum, Vol. 108, No. 1 (2007), pp. 31-51.

Crossrefs

Cf. A000010, A000203, A033632, A062401, A062402, A065395, A066930, A289336, A073858 (positions of 1's), A289412, A370690 (denominators).

Cf. A080130, A091724.

Programs

Mathematica

Table[DivisorSigma[1, EulerPhi[n]]/EulerPhi[DivisorSigma[1, n]], {n, 1, 100}] // Numerator

PARI

a(n) = {my(f = factor(n)); numerator(sigma(eulerphi(f)) / eulerphi(sigma(f)));}

Formula

Let f(n) = a(n)/A370690(n) = A062402(n)/A062401(n).

Formulas from De Koninck and Luca (2007):

lim sup_{n->oo} f(n)/log_2(n)^2 = exp(2*gamma) (A091724).

lim inf_{n->oo} f(n)/log_2(n)^2 = delta exists, and exp(-gamma)/40 <= delta <= 2*exp(-gamma).

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... .