A366442 The sum of divisors of the 5-rough numbers (A007310).

1, 6, 8, 12, 14, 18, 20, 24, 31, 30, 32, 48, 38, 42, 44, 48, 57, 54, 72, 60, 62, 84, 68, 72, 74, 96, 80, 84, 108, 90, 112, 120, 98, 102, 104, 108, 110, 114, 144, 144, 133, 156, 128, 132, 160, 138, 140, 168, 180, 150, 152, 192, 158, 192, 164, 168, 183, 174, 248

Offset: 1

Author

Amiram Eldar, Oct 10 2023

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A000203, A013661, A007310, A100044, A366441.

Similar sequences: A008438, A062822, A065764, A180114, A362986, A366439, A366440.

Programs

Mathematica

a[n_] := DivisorSigma[1, 2*Floor[3*n/2] - 1]; Array[a, 100]

PARI

a(n) = sigma((3*n)\2 << 1 - 1)

Python

from sympy import divisor_sigma
def A366442(n): return divisor_sigma((n+(n>>1)<<1)-1) # Chai Wah Wu, Oct 10 2023

Formula

a(n) = A000203(A007310(n)).

Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2) = 1.644934... (A013661).

The asymptotic mean of the abundancy index of the 5-rough numbers: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A007310(k) = Pi^2/9 = 1.0966227... (A100044).

In general, the asymptotic mean of the abundancy index of the prime(k)-rough numbers is zeta(2) * Product_{i=1..k-1} (1 - 1/prime(i)^2).