A374979 a(n) = Sum_{i+j+k+l+m+r=n, i,j,k,l,m,r >= 1} sigma(i)*sigma(j)*sigma(k)*sigma(l)*sigma(m)*sigma(r).

0, 0, 0, 0, 0, 1, 18, 159, 942, 4281, 16050, 51932, 149532, 391524, 947246, 2143677, 4581204, 9316195, 18138636, 33984912, 61534652, 108055425, 184582014, 307515038, 500798058, 798762453, 1249917936, 1921788036, 2907159804, 4332046200, 6365441400, 9232216725

Offset: 1

Author

Chai Wah Wu, Jul 26 2024

Keywords

• nonn

Comments

6-fold convolution of A000203.

Convolution of A000203 and A374978.

a(n) = Sum_{i=1..n-1} A000203(i)*A374978(n-i).

a(n) = Sum_{i=1..n-2} A000385(i)*A374977(n-i-1).

a(n) = Sum_{i=1..n-1} A374951(i)*A374951(n-i).

a(n) = Sum_{i+j+k=n-3, i,j,k>=1} A000385(i)*A000385(j)*A000385(k).

Column k=6 of A319083.

In general, if the sequence "a" is a k-fold convolution of A000203, then Sum_{k=1..n} a(k) ~ Pi^(2*k) * n^(2*k) / (6^k * (2*k)!). - Vaclav Kotesovec, Sep 20 2024

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000385, A319083, A374951, A374977, A374978.

Programs

Maple

b:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
       add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 6):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 26 2024

Python

from functools import lru_cache
from sympy import divisor_sigma
def A374979(n):
    @lru_cache(maxsize=None)
    def g(x):
        f = factorint(x+1).items()
        return(5*prod((p**(3*(e+1))-1)//(p**3-1) for p,e in f)-(5+6*x)*prod((p**(e+1)-1)//(p-1) for p, e in f))//12
    return sum(g(i)*g(j)*g(n-3-i-j) for i in range(1,n-4) for j in range(1,n-i-3))

Formula

Sum_{k=1..n} a(k) ~ Pi^12 * n^12 / 22348298649600. - Vaclav Kotesovec, Sep 20 2024