A366970 -id:A366970 - 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

A366968 a(n) = Sum_{k=3..n} floor(n/k).

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 6, 8, 10, 12, 13, 17, 18, 20, 23, 26, 27, 31, 32, 36, 39, 41, 42, 48, 50, 52, 55, 59, 60, 66, 67, 71, 74, 76, 79, 86, 87, 89, 92, 98, 99, 105, 106, 110, 115, 117, 118, 126, 128, 132, 135, 139, 140, 146, 149, 155, 158, 160, 161, 171, 172, 174, 179, 184

Offset: 1

Seiichi Manyama, Oct 30 2023

Column k=3 of A134867.
Partial sums of A023645.
Cf. A006218, A366969, A366970, A366971.

PARI
```
a(n) = sum(k=3, n, n\k);
```

from math import isqrt
def A366968(n): return -(s:=isqrt(n))**2+(sum(n//k for k in range(3,s+1))<<1)+n+(n>>1) if n>3 else int(n>2) # Chai Wah Wu, Oct 30 2023

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

a(n) = A006218(n)-n-floor(n/2). - Chai Wah Wu, Oct 30 2023

A366969 a(n) = Sum_{k=3..n} (k-2) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 16, 24, 32, 43, 52, 69, 80, 97, 114, 136, 151, 179, 196, 227, 252, 281, 302, 347, 373, 408, 441, 486, 513, 570, 599, 651, 692, 739, 780, 854, 889, 942, 991, 1066, 1105, 1186, 1227, 1300, 1367, 1432, 1477, 1582, 1634, 1716, 1781, 1868, 1919, 2024

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A152771.

Cf. A006218, A024916, A366968, A366970, A366971.

PARI
```
a(n) = sum(k=3, n, (k-2)*(n\k));
```

from math import isqrt
def A366969(n): return n+(-(s:=isqrt(n))*(s*(s-2)-7)+sum(((q:=n//w)+1)*(q+(w<<1)-8) for w in range(1,s+1))>>1) # Chai Wah Wu, Oct 30 2023

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

a(n) = n + A024916(n) - 2*A006218(n). - Chai Wah Wu, Oct 30 2023

cp's OEIS Frontend

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

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PARI

Python

Formula

A366968 a(n) = Sum_{k=3..n} floor(n/k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

A366969 a(n) = Sum_{k=3..n} (k-2) * floor(n/k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula