A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

1, 6, 17, 42, 78, 149, 234, 379, 555, 815, 1102, 1557, 2013, 2662, 3388, 4349, 5319, 6695, 8026, 9846, 11712, 14027, 16328, 19503, 22464, 26200, 30030, 34759, 39255, 45221, 50678, 57623, 64465, 72579, 80469, 90665, 99805, 111020, 122146, 135566, 147908, 163638

Offset: 1

Author

Seiichi Manyama, Oct 23 2023

Keywords

• nonn
• easy

Crossrefs

Partial sums of A059358.

Cf. A006218, A024916, A064602, A064603, A364970, A365439.

Programs

PARI

a(n) = sum(k=1, n, binomial(n\k+3, 4));

Python

from math import isqrt, comb
def A365409(n): return -(s:=isqrt(n))**2*comb(s+3,3)+sum((q:=n//k)*((comb(k+2,3)<<2)+comb(q+3,3)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 26 2023

Formula

a(n) = Sum_{k=1..n} binomial(k+2,3) * floor(n/k).

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1-x^k).

a(n) = (A064603(n)+3*A064602(n)+2*A024916(n))/6. - Chai Wah Wu, Oct 26 2023