A366967 -id:A366967 - cp's OEIS frontend

A366971 a(n) = Sum_{k=3..n} binomial(k,3) * floor(n/k).

0, 0, 1, 5, 15, 36, 71, 131, 216, 346, 511, 756, 1042, 1441, 1907, 2527, 3207, 4128, 5097, 6371, 7737, 9442, 11213, 13538, 15848, 18734, 21744, 25423, 29077, 33743, 38238, 43818, 49440, 56104, 62694, 70979, 78749, 88154, 97580, 108790, 119450, 132680, 145021, 159974

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363607.
Cf. A366968, A366969, A366970.
Cf. A024916, A064602, A064603, A366967.

a(n) = sum(k=3, n, binomial(k, 3)*(n\k));

from math import isqrt, comb
def A366971(n): return -comb((s:=isqrt(n))+1,4)*(s+1)+sum(comb((q:=n//w)+1,4)+(q+1)*comb(w,3) for w in range(1,s+1)) # Chai Wah Wu, Oct 30 2023

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

a(n) = (A064603(n) - 3*A064602(n) + 2*A024916(n))/6. - Chai Wah Wu, Oct 30 2023

A366984 a(n) = Sum_{k=1..n} binomial(k+2,2) * floor(n/k).

Original entry on oeis.org

3, 12, 25, 49, 73, 120, 159, 228, 296, 392, 473, 626, 734, 899, 1069, 1291, 1465, 1757, 1970, 2312, 2614, 2977, 3280, 3803, 4178, 4670, 5144, 5759, 6227, 6993, 7524, 8307, 8993, 9803, 10529, 11630, 12374, 13373, 14311, 15559, 16465, 17867, 18860, 20273, 21579, 23016

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363628.
Cf. A006218, A366983, A366985.
Cf. A366967.

a(n) = sum(k=1, n, binomial(k+2, 2)*(n\k));

from math import isqrt
def A366984(n): return (-(s:=isqrt(n))*(s*(s*(s+7)+17)+17)+sum(((q:=n//w)+1)*(q*(q+5)+3*(w*(w+3)+4)) for w in range(1,s+1)))//6 # Chai Wah Wu, Oct 31 2023

G.f.: 1/(1-x) * Sum_{k>0} (1/(1-x^k)^3 - 1) = 1/(1-x) * Sum_{k>0} binomial(k+2,2) * x^k/(1-x^k).

cp's OEIS Frontend

A366971 a(n) = Sum_{k=3..n} binomial(k,3) * floor(n/k).

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