A236632 -id:A236632 - cp's OEIS frontend

A366967 a(n) = Sum_{k=2..n} binomial(k,2) * floor(n/k).

0, 1, 4, 11, 21, 40, 61, 96, 135, 191, 246, 337, 415, 528, 646, 801, 937, 1145, 1316, 1568, 1802, 2089, 2342, 2737, 3047, 3451, 3841, 4338, 4744, 5358, 5823, 6474, 7060, 7758, 8384, 9294, 9960, 10835, 11657, 12717, 13537, 14739, 15642, 16881, 18025, 19314, 20395

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A069153.
Cf. A002541, A236632.
Cf. A024916, A064602, A366971.

a(n) = sum(k=2, n, binomial(k, 2)*(n\k));

from math import isqrt
def A366967(n): return ((s:=isqrt(n))**2*(1-s**2)+sum((q:=n//k)*(3*k*(k-1)+q**2-1) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

G.f.: 1/(1-x) * Sum_{k>=1} x^(2*k)/(1-x^k)^3 = 1/(1-x) * Sum_{k>=2} binomial(k,2) * x^k/(1-x^k).

a(n) = (A064602(n)-A024916(n))/2. - Chai Wah Wu, Oct 30 2023

A356100 a(n) = Sum_{k=1..n} (k - 1)^n * floor(n/k).

Original entry on oeis.org

0, 1, 9, 99, 1301, 20581, 376891, 7914216, 186905206, 4915451602, 142368695176, 4506118905870, 154720069309364, 5729167232515112, 227585086051159866, 9654819212943764500, 435659280972794395356, 20836049921760968809231, 1052864549462731148832219

Offset: 1

Seiichi Manyama, Jul 26 2022

nonn

Cf. A121706, A236632, A319194, A332469.

Table[Sum[(k-1)^n Floor[n/k],{k,n}],{n,20}] (* Harvey P. Dale, Dec 14 2024 *)

PARI
```
a(n) = sum(k=1, n, (k-1)^n*(n\k));
```

a(n) = sum(k=1, n, sigma(k, n)-(n\k)^n);

a(n) = sum(k=1, n, sumdiv(k, d, (d-1)^n));

def A356100(n): return sum((k-1)**n*(n//k) for k in range(2,n+1)) # Chai Wah Wu, Jul 26 2022

a(n) = A319194(n) - A332469(n).
a(n) = Sum_{k=1..n} Sum_{d|k} (d - 1)^n.
a(n) = [x^n] (1/(1-x)) * Sum_{k>=1} (k - 1)^n * x^k/(1 - x^k).

cp's OEIS Frontend

A366967 a(n) = Sum_{k=2..n} binomial(k,2) * floor(n/k).

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A356100 a(n) = Sum_{k=1..n} (k - 1)^n * floor(n/k).

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