A289336 a(n) = numerator of (sigma(n) / phi(n)).

1, 3, 2, 7, 3, 6, 4, 15, 13, 9, 6, 7, 7, 4, 3, 31, 9, 13, 10, 21, 8, 18, 12, 15, 31, 7, 20, 14, 15, 9, 16, 63, 12, 27, 2, 91, 19, 10, 7, 45, 21, 8, 22, 21, 13, 36, 24, 31, 19, 93, 9, 49, 27, 20, 9, 5, 20, 45, 30, 21, 31, 16, 26, 127, 7, 36, 34, 63, 24, 6, 36

Offset: 1

Author

Jaroslav Krizek, Aug 19 2017

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with: 1, 3, 2, 7/2, 3/2, 6, 4/3, 15/4, 13/6, 9/2, 6/5, 7, ...
For n = 7, sigma(7) / phi(7) = 8/6 = 4/3, a(7) = 4.
		

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. A289412 (denominators).

Cf. A000010, A000203, A018894, A020492, A023897, A055234, A070032, A070033.

Programs

Magma

[Numerator(SumOfDivisors(n) / EulerPhi(n)): n in[1..1000]]

Mathematica

Array[Numerator[DivisorSigma[1, #]/EulerPhi[#]] &, 71] (* Michael De Vlieger, Aug 19 2017 *)

PARI

a(n) = numerator(sigma(n)/eulerphi(n)); \\ Michel Marcus, Aug 21 2017

Formula

a(n) = numerator of (A000203(n) / A000010(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A289412(k) = (Pi^4/36) * Product_{p prime} (1 + 2/p^3 - 1/p^5) = 3.6174451656... . - Amiram Eldar, Nov 21 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} A289412(k)/a(k) = 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... (De Koninck and Luca, 2007). - Amiram Eldar, Feb 27 2024