A370690 -id:A370690 - cp's OEIS frontend

A091724 Decimal expansion of e^(2*EulerGamma).

3, 1, 7, 2, 2, 1, 8, 9, 5, 8, 1, 2, 5, 4, 5, 0, 5, 2, 7, 7, 2, 7, 9, 1, 3, 4, 0, 9, 0, 6, 9, 4, 7, 4, 9, 7, 7, 1, 2, 2, 9, 5, 7, 7, 3, 7, 7, 7, 2, 3, 0, 0, 4, 5, 8, 5, 1, 4, 7, 7, 8, 2, 8, 8, 4, 1, 9, 2, 5, 2, 1, 4, 4, 1, 1, 6, 3, 8, 9, 4, 6, 3, 6, 6, 4, 6, 3, 8, 1, 7, 8, 7, 5, 0, 8, 4, 8, 9, 6, 6, 6, 5

Offset: 1

Eric W. Weisstein, Feb 01 2004

			3.17221895812545052772791340906947497712295773777230...

Robert Price, 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.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A001620, A073004, A370689, A370690.

RealDigits[Exp[2*EulerGamma], 10, 100][[1]] (* Amiram Eldar, Jun 25 2021 *)

exp(2*Euler) \\ Michel Marcus, Jun 25 2021

Equals lim_{x -> 0} e^(2*ExpIntegralEi(-x))/x^2.
Equals A073004^2. - Michel Marcus, Jun 25 2021
Equals lim sup_{n->oo} H(n)/log_2(n)^2, where H(n) = A370689(n)/A370690(n) (De Koninck and Luca, 2007). - Amiram Eldar, Feb 27 2024

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.

Original entry on oeis.org

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

Amiram Eldar, Feb 27 2024

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

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.

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

Cf. A080130, A091724.

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

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

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... .

cp's OEIS Frontend

A091724 Decimal expansion of e^(2*EulerGamma).

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