A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

1, 7, 23, 64, 135, 282, 493, 864, 1375, 2166, 3168, 4715, 6536, 9132, 12278, 16525, 21371, 27998, 35314, 44995, 55847, 69504, 84455, 103882, 124428, 150005, 177921, 212017, 247978, 292890, 339267, 395874, 455796, 526692, 600788, 691066, 782457, 891048, 1004814

Offset: 1

Author

Seiichi Manyama, Oct 23 2023

Keywords

• nonn
• easy

Crossrefs

Partial sums of A073570.

Cf. A006218, A024916, A064602, A064603, A064604, A364970, A365409.

Programs

PARI

a(n) = sum(k=1, n, binomial(n\k+4, 5));

Python

from math import isqrt, comb
def A365439(n): return (-(s:=isqrt(n))**2*comb(s+4,4)+sum((q:=n//k)*(5*comb(k+3,4)+comb(q+4,4)) for k in range(1,s+1)))//5 # Chai Wah Wu, Oct 26 2023

Formula

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

G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^5 = 1/(1-x) * Sum_{k>=1} binomial(k+3,4) * x^k/(1-x^k).

a(n) = (A064604(n)+6*A064603(n)+11*A064602(n)+6*A024916(n))/24. - Chai Wah Wu, Oct 26 2023