A386401 a(n) = numerator(sigma(n)*phi(n)/n^2).

1, 3, 8, 7, 24, 2, 48, 15, 26, 18, 120, 7, 168, 36, 64, 31, 288, 13, 360, 21, 128, 90, 528, 5, 124, 126, 80, 6, 840, 16, 960, 63, 320, 216, 1152, 91, 1368, 270, 448, 9, 1680, 32, 1848, 105, 208, 396, 2208, 31, 342, 93, 256, 147, 2808, 20, 576, 45, 320, 630, 3480, 56

Offset: 1

Author

Stefano Spezia, Jul 20 2025

Keywords

• nonn
• easy
• frac

Comments

a(n)/A386402(n) = sigma(n)*phi(n)*(1/n^2) is a multiplicative function since it is the product of three multiplicative functions.

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, Exercise 5.3.21 on page 169.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000203, A000290, A062354, A065465.

Cf. A386402 (denominators).

Programs

Mathematica

a[n_]:=Numerator[DivisorSigma[1,n]EulerPhi[n]/n^2]; Array[a,60]

PARI

a(n) = {my(f = factor(n)); numerator(sigma(f) * eulerphi(f) / n^2);} \\ Amiram Eldar, Jul 21 2025

Formula

From Amiram Eldar, Jul 21 2025: (Start)

Let f(n) = a(n)/A386402(n) = sigma(n)*phi(n)/n^2. Then:

f(n) = A062354(n)/n^2.

f(n) is multiplicative with f(p^e) = 1 - 1/p^(e+1).

Dirichlet g.f. of f(n): zeta(s) * zeta(s+1) * Product_{p prime} (1 - 1/p^(s+1) - 1/p^(s+2)+ 1/p^(2*s+2)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} (1 - 1/(p^2*(p+1))) = 0.881513... (A065465). (End)