A366971 -id:A366971 - cp's OEIS frontend

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

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

A366970 a(n) = Sum_{k=3..n} binomial(k-1,2) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 36, 60, 89, 131, 176, 245, 311, 404, 502, 631, 751, 926, 1079, 1295, 1501, 1756, 1987, 2330, 2612, 2978, 3332, 3779, 4157, 4707, 5142, 5736, 6278, 6926, 7508, 8336, 8966, 9785, 10555, 11533, 12313, 13427, 14288, 15449, 16521, 17742, 18777, 20306

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A363610.

Cf. A006218, A024916, A064602, A366968, A366969, A366971.

a(n) = sum(k=3, n, binomial(k-1, 2)*(n\k));

from math import isqrt
def A366970(n): return (-(s:=isqrt(n))*(s*(s**2-(s<<1)-1)+8)+sum(((q:=n//w)+1)*(q*(q-4)+3*(w**2-3*w+4)) for w in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

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

a(n) = (A064602(n)-3*A024916(n))/2 + A006218(n). - Chai Wah Wu, Oct 30 2023

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

Original entry on oeis.org

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

A366985 a(n) = Sum_{k=1..n} binomial(k+3,3) * floor(n/k).

Original entry on oeis.org

4, 18, 42, 91, 151, 269, 393, 607, 851, 1207, 1575, 2183, 2747, 3561, 4457, 5640, 6784, 8452, 9996, 12158, 14326, 17004, 19608, 23306, 26642, 30870, 35174, 40518, 45482, 52214, 58202, 65930, 73458, 82382, 90998, 102295, 112179, 124393, 136457, 151125, 164373

Offset: 1

Seiichi Manyama, Oct 30 2023

nonn

Partial sums of A116963.
Cf. A006218, A366983, A366984.
Cf. A366971.

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

from math import isqrt
def A366985(n): return (-(s:=isqrt(n))*(s*(s*(s*(s+11)+45)+85)+74)+sum(((q:=n//w)+1)*(q*(q*(q+9)+26)+((w+4)*(w*(w+2)+3)<<2)) for w in range(1,s+1)))//3>>3 # Chai Wah Wu, Oct 31 2023

G.f.: 1/(1-x) * Sum_{k>0} (1/(1-x^k)^4 - 1) = 1/(1-x) * Sum_{k>0} binomial(k+3,3) * x^k/(1-x^k).

cp's OEIS Frontend

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

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PARI

Python

Formula

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

A366970 a(n) = Sum_{k=3..n} binomial(k-1,2) * floor(n/k).

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Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

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

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula

A366985 a(n) = Sum_{k=1..n} binomial(k+3,3) * floor(n/k).

Views

Author

Keywords

Crossrefs

Programs

PARI

Python

Formula