A268235 -id:A268235 - cp's OEIS frontend

A059851 a(n) = n - floor(n/2) + floor(n/3) - floor(n/4) + ... (this is a finite sum).

0, 1, 1, 3, 2, 4, 4, 6, 4, 7, 7, 9, 7, 9, 9, 13, 10, 12, 12, 14, 12, 16, 16, 18, 14, 17, 17, 21, 19, 21, 21, 23, 19, 23, 23, 27, 24, 26, 26, 30, 26, 28, 28, 30, 28, 34, 34, 36, 30, 33, 33, 37, 35, 37, 37, 41, 37, 41, 41, 43, 39, 41, 41, 47, 42, 46, 46, 48, 46, 50, 50, 52, 46, 48, 48

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Feb 27 2001

As n goes to infinity we have the asymptotic formula: a(n) ~ n * log(2).

More precisely, a(n) = n * log(2) + O(n^(131/416) * (log n)^(26947/8320)). - V Sai Prabhav, Jun 02 2025

			a(5) = 4 because floor(5) - floor(5/2) + floor(5/3) - floor(5/4) + floor(5/5) - floor(5/6) + ... = 5 - 2 + 1 - 1 + 1 - 0 + 0 - 0 + ... = 4.

T. D. Noe, Table of n, a(n) for n = 0..10000
V Sai Prabhav, Asymptotic Expansion of a(n)

Cf. A000005, A075997, A271860, A263086.
Partial sums of A048272.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), this sequence (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A059851:= func< n | (&+[Floor(n/j)*(-1)^(j-1): j in [1..n]]) >;
[A059851(n): n in [1..80]]; // G. C. Greubel, Jun 27 2024

for n from 0 to 200 do printf(`%d,`, sum((-1)^(i+1)*floor(n/i), i=1..n)) od:

f[list_, i_] := list[[i]]; nn = 200; a = Table[1, {n, 1, nn}]; b =
Table[If[OddQ[n], 1, -1], {n, 1, nn}];Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] // Accumulate (* Geoffrey Critzer, Mar 29 2015 *)
Table[Sum[Floor[n/k] - 2*Floor[n/(2*k)], {k, 1, n}], {n, 0, 100}] (* Vaclav Kotesovec, Dec 23 2020 *)

{ for (n=0, 10000, s=1; d=2; a=n; while ((f=floor(n/d)) > 0, a-=s*f; s=-s; d++); write("b059851.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 29 2009

from math import isqrt
def A059851(n): return ((t:=isqrt(m:=n>>1))**2<<1)-(s:=isqrt(n))**2+(sum(n//k for k in range(1,s+1))-(sum(m//k for k in range(1,t+1))<<1)<<1) # Chai Wah Wu, Oct 23 2023

def A059851(n): return sum((n//j)*(-1)^(j-1) for j in range(1,n+1))
[A059851(n) for n in range(81)] # G. C. Greubel, Jun 27 2024

From Vladeta Jovovic, Oct 15 2002: (Start)
a(n) = A006218(n) - 2*A006218(floor(n/2)).
G.f.: 1/(1-x)*Sum_{n>=1} x^n/(1+x^n). (End)
a(n) = Sum_{n/2 < k < =n} d(k) - Sum_{1 < =k <= n/2} d(k), where d(k) = A000005(k). Also, a(n) = number of terms among {floor(n/k)}, 1<=k<=n, that are odd. - Leroy Quet, Jan 19 2006
From Ridouane Oudra, Aug 15 2019: (Start)
a(n) = Sum_{k=1..n} (floor(n/k) mod 2).
a(n) = (1/2)*(n + A271860(n)).
a(n) =  Sum_{k=1..n} round(n/(2*k)) - floor(n/(2*k)), where round(1/2) = 1. (End)
a(n) = 2*A263086(n) - 3*A006218(n). - Ridouane Oudra, Aug 17 2024

More terms from James Sellers and Larry Reeves (larryr(AT)acm.org), Feb 27 2001

A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

Original entry on oeis.org

1, 4, 15, 92, 790, 9384, 137326, 2397352, 48428487, 1111122360, 28531183329, 810554859732, 25239592620853, 854769763924104, 31278135039463245, 1229782938533709200, 51702516368332126932, 2314494592676172411516, 109912203092257573556274, 5518821052632117898282620

Offset: 1

Ilya Gutkovskiy, Feb 16 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A024916, A268235, A308313, A308814, A319194, A320095.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), this sequence (q=n).

A332533:= func< n | (&+[Floor(n/j)*n^(j-1): j in [1..n]]) >;
[A332533(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024

seq(add(n^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023

Table[(1/n) Sum[Floor[n/k] n^k, {k, 1, n}], {n, 1, 20}]
Table[(1/n) Sum[Sum[n^d, {d, Divisors[k]}], {k, 1, n}], {n, 1, 20}]
Table[SeriesCoefficient[(1/(1 - x)) Sum[x^k/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 1, 20}]

a(n) = sum(k=1, n, (n\k)*n^k)/n; \\ Michel Marcus, Feb 16 2020

a(n) = sum(k=1, n, sumdiv(k, d, n^(d-1))); \\ Seiichi Manyama, May 29 2021

def A332533(n): return sum((n//j)*n^(j-1) for j in range(1,n+1))
[A332533(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k / (1 - n*x^k).
a(n) = (1/n) * Sum_{k=1..n} Sum_{d|k} n^d.
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 28 2021
a(n) = (1/(n-1)) * Sum_{k=1..n} (n^floor(n/k) - 1), for n>=2. - Ridouane Oudra, Mar 05 2023

A344814 a(n) = Sum_{k=1..n} floor(n/k) * 3^(k-1).

Original entry on oeis.org

1, 5, 15, 46, 128, 384, 1114, 3332, 9903, 29671, 88721, 266151, 797593, 2392649, 7175709, 21526834, 64573556, 193720536, 581141026, 1743422288, 5230207428, 15690619684, 47071679294, 141215037738, 423644574301, 1270933715189, 3812799550089, 11438398630159

Offset: 1

Seiichi Manyama, May 29 2021

nonn

Partial sums of A034730.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=3 of A344821.

Cf. A059851, A268235, A332533, A344815, A344816, A344817, A344818, A344819, A344820.

A344814:= func< n | (&+[Floor(n/k)*3^(k-1): k in [1..n]]) >;
[A344814(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

seq(add(3^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Feb 05 2023

a[n_] := Sum[3^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*3^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, 3^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-3*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 3^(k-1)*x^k/(1-x^k))/(1-x))

def A344814(n): return sum((n//k)*3^(k-1) for k in range(1,n+1))
[A344814(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} 3^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 3*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 3^(k-1) * x^k/(1 - x^k).
a(n) ~ 3^n / 2. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/2) * Sum_{k=1..n} (3^floor(n/k) - 1). - Ridouane Oudra, Feb 05 2023

A344815 a(n) = Sum_{k=1..n} floor(n/k) * 4^(k-1).

Original entry on oeis.org

1, 6, 23, 92, 349, 1394, 5491, 21944, 87497, 349902, 1398479, 5593892, 22371109, 89484074, 357919803, 1431678080, 5726645377, 22906581142, 91626057879, 366504227292, 1466015859181, 5864063418866, 23456249463283, 93824997852744, 375299974563657, 1501199898183502

Offset: 1

Seiichi Manyama, May 29 2021

nonn

Partial sums of A339684.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=4 of A344821.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), this sequence (q=4), A344816 (q=5), A332533 (q=n).

A344815:= func< n | (&+[Floor(n/j)*4^(j-1): j in [1..n]]) >;
[A344815(n): n in [1..40]]; // G. C. Greubel, Jun 27 2024

seq(add(4^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Feb 16 2023

a[n_] := Sum[4^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*4^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, 4^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-4*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 4^(k-1)*x^k/(1-x^k))/(1-x))

def A344815(n): return sum((n//j)*4^(j-1) for j in range(1,n+1))
[A344815(n) for n in range(1,41)] # G. C. Greubel, Jun 27 2024

a(n) = Sum_{k=1..n} Sum_{d|k} 4^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 4*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 4^(k-1) * x^k/(1 - x^k).
a(n) ~ 4^n / 3. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/3) * Sum_{k=1..n} (4^floor(n/k) - 1). - Ridouane Oudra, Feb 16 2023

A344816 a(n) = Sum_{k=1..n} floor(n/k) * 5^(k-1).

Original entry on oeis.org

1, 7, 33, 164, 790, 3946, 19572, 97828, 488479, 2442235, 12207861, 61039267, 305179893, 1525898649, 7629414925, 38147071306, 190734961932, 953674808838, 4768372074464, 23841860356470, 119209292012746, 596046459981502, 2980232250997128, 14901161254984784

Offset: 1

Seiichi Manyama, May 29 2021

nonn

Partial sums of A339685.

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=5 of A344821.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), this sequence (q=5), A332533 (q=n).

A344816:= func< n | (&+[Floor(n/k)*5^(k-1): k in [1..n]]) >;
[A344816(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

seq(add(5^(k-1)*floor(n/k), k=1..n), n=1..60); # Ridouane Oudra, Mar 05 2023

a[n_] := Sum[5^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*5^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, 5^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-5*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, 5^(k-1)*x^k/(1-x^k))/(1-x))

def A344816(n): return sum((n//k)*5^(k-1) for k in range(1,n+1))
[A344816(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} 5^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 - 5*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} 5^(k-1) * x^k/(1 - x^k).
a(n) ~ 5^n / 4. - Vaclav Kotesovec, Jun 05 2021
a(n) = (1/4) * Sum_{k=1..n} (5^floor(n/k) - 1). - Ridouane Oudra, Mar 05 2023

A344817 a(n) = Sum_{k=1..n} floor(n/k) * (-2)^(k-1).

Original entry on oeis.org

1, 0, 5, -4, 13, -16, 49, -88, 173, -324, 701, -1384, 2713, -5416, 10989, -21916, 43621, -87224, 174921, -349872, 698773, -1397356, 2796949, -5593872, 11183361, -22366976, 44742149, -89483716, 178951741, -357903312, 715838513, -1431678040, 2863290285

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=2 of A344824.
Cf. A081295.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), A344818 (q=-3), this sequence (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A344817:= func< n | (&+[Floor(n/k)*(-2)^(k-1): k in [1..n]]) >;
[A344817(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

a[n_] := Sum[(-2)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-2)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-2)^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+2*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-2)^(k-1)*x^k/(1-x^k))/(1-x))

def A344817(n): return sum((n//k)*(-2)^(k-1) for k in range(1,n+1))
[A344817(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-2)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 2*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-2)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 2^n / 3. - Vaclav Kotesovec, Jun 05 2021

A344820 a(n) = Sum_{k=1..n} floor(n/k) * (-n)^(k-1).

Original entry on oeis.org

1, 0, 9, -52, 520, -6636, 102984, -1864600, 38741463, -909081740, 23775986069, -685854111804, 21633935838489, -740800448012044, 27368368159530285, -1085102592823737200, 45957792326631241516, -2070863582899905915336, 98920982783031811482920, -4993219047619535240997780

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..387

Diagonal of A344824.

Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): this sequence (q=-n), A344819 (q=-4), A344818 (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A344820:= func< n | (&+[Floor(n/k)*(-n)^(k-1): k in [1..n]]) >;
[A344820(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

a[n_] := Sum[(-n)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 20] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-n)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-n)^(d-1)));

def A344820(n): return sum((n//k)*(-n)^(k-1) for k in range(1,n+1))
[A344820(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-n)^(d-1).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} x^k/(1 + n*x^k).
a(n) = [x^n] (1/(1 - x)) * Sum_{k>=1} (-n)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * n^(n-1). - Vaclav Kotesovec, Jun 05 2021

A344818 a(n) = Sum_{k=1..n} floor(n/k) * (-3)^(k-1).

Original entry on oeis.org

1, -1, 9, -20, 62, -174, 556, -1660, 4911, -14693, 44357, -133053, 398389, -1195207, 3587853, -10763270, 32283452, -96850386, 290570104, -871710994, 2615074146, -7845220010, 23535839600, -70607518824, 211822017739, -635466060265, 1906399774635, -5719199303975

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=3 of A344824.
Cf. A101561.
Sums of the form Sum_{k=1..n} q^(k-1)*floor(n/k): A344820 (q=-n), A344819 (q=-4), this sequence (q=-3), A344817 (q=-2), A059851 (q=-1), A006218 (q=1), A268235 (q=2), A344814 (q=3), A344815 (q=4), A344816 (q=5), A332533 (q=n).

A344818:= func< n | (&+[Floor(n/k)*(-3)^(k-1): k in [1..n]]) >;
[A344818(n): n in [1..40]]; // G. C. Greubel, Jun 26 2024

a[n_] := Sum[(-3)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-3)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-3)^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+3*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-3)^(k-1)*x^k/(1-x^k))/(1-x))

def A344818(n): return sum((n//k)*(-3)^(k-1) for k in range(1,n+1))
[A344818(n) for n in range(1,41)] # G. C. Greubel, Jun 26 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-3)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 3*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-3)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 3^n / 4. - Vaclav Kotesovec, Jun 05 2021

A344819 a(n) = Sum_{k=1..n} floor(n/k) * (-4)^(k-1).

Original entry on oeis.org

1, -2, 15, -52, 205, -806, 3291, -13160, 52393, -209498, 839079, -3356300, 13420917, -53683854, 214751875, -859006400, 3435960897, -13743843762, 54975632975, -219902535924, 879609095965, -3518436366566, 14073749677851, -56294998711576, 225179977999337, -900719912066074

Offset: 1

Seiichi Manyama, May 29 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Column k=4 of A344824.

Cf. A059851, A101562, A268235, A332533, A344814, A344815, A344816, A344817, A344818, A344820.

A344819:= func< n | (&+[(-4)^(k-1)*Floor(n/k): k in [1..n]]) >;
[A344819(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

a[n_] := Sum[(-4)^(k - 1) * Quotient[n, k], {k, 1, n}]; Array[a, 30] (* Amiram Eldar, May 29 2021 *)

PARI
```
a(n) = sum(k=1, n, n\k*(-4)^(k-1));
```

a(n) = sum(k=1, n, sumdiv(k, d, (-4)^(d-1)));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1+4*x^k))/(1-x))

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (-4)^(k-1)*x^k/(1-x^k))/(1-x))

def A344819(n): return sum((-4)^(k-1)*int(n//k) for k in range(1,n+1))
[A344819(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

a(n) = Sum_{k=1..n} Sum_{d|k} (-4)^(d-1).
G.f.: (1/(1 - x)) * Sum_{k>=1} x^k/(1 + 4*x^k).
G.f.: (1/(1 - x)) * Sum_{k>=1} (-4)^(k-1) * x^k/(1 - x^k).
a(n) ~ -(-1)^n * 4^n / 5. - Vaclav Kotesovec, Jun 05 2021

A344821 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{j=1..n} floor(n/j) * k^(j-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 9, 8, 5, 1, 6, 15, 20, 10, 6, 1, 7, 23, 46, 37, 14, 7, 1, 8, 33, 92, 128, 76, 16, 8, 1, 9, 45, 164, 349, 384, 141, 20, 9, 1, 10, 59, 268, 790, 1394, 1114, 280, 23, 10, 1, 11, 75, 410, 1565, 3946, 5491, 3332, 541, 27, 11

Offset: 1

Seiichi Manyama, May 29 2021

			Square array, A(n, k), begins:
  1,  1,  1,   1,    1,    1,    1, ...
  2,  3,  4,   5,    6,    7,    8, ...
  3,  5,  9,  15,   23,   33,   45, ...
  4,  8, 20,  46,   92,  164,  268, ...
  5, 10, 37, 128,  349,  790, 1565, ...
  6, 14, 76, 384, 1394, 3946, 9384, ...
Antidiagonal triangle, T(n, k), begins:
  1;
  1,  2;
  1,  3,  3;
  1,  4,  5,   4;
  1,  5,  9,   8,   5;
  1,  6, 15,  20,  10,   6;
  1,  7, 23,  46,  37,  14,   7;
  1,  8, 33,  92, 128,  76,  16,  8;
  1,  9, 45, 164, 349, 384, 141, 20,  9;

G. C. Greubel, Antidiagonals n = 1..50, flattened

Columns k=0..5 give A000027, A006218, A268235, A344814, A344815, A344816.
A(n,n) gives A332533.
Cf. A308813, A344824.

A:= func< n,k | k eq n select n else (&+[Floor(n/j)*k^(j-1): j in [1..n]]) >;
A344821:= func< n,k | A(k+1, n-k-1) >;
[A344821(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 27 2024

A[n_, k_] := Sum[If[k == 0 && j == 1, 1, k^(j - 1)] * Quotient[n, j], {j, 1, n}]; Table[A[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 29 2021 *)

PARI
```
A(n, k) = sum(j=1, n, n\j*k^(j-1));
```

A(n, k) = sum(j=1, n, sumdiv(j, d, k^(d-1)));

def A(n,k): return n if k==n else sum((n//j)*k^(j-1) for j in range(1,n+1))
def A344821(n,k): return A(k+1, n-k-1)
flatten([[A344821(n,k) for k in range(n)] for n in range(1,13)]) # G. C. Greubel, Jun 27 2024

G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j/(1 - k*x^j).
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} k^(j-1) * x^j/(1 - x^j).
A(n, k) = Sum_{j=1..n} Sum_{d|j} k^(d - 1).
T(n, k) = Sum_{j=1..k+1} floor((k+1)/j) * (n-k-1)^(j-1), for n >= 1, 0 <= k <= n-1 (antidiagonal triangle). - G. C. Greubel, Jun 27 2024

cp's OEIS Frontend

A059851 a(n) = n - floor(n/2) + floor(n/3) - floor(n/4) + ... (this is a finite sum).

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A332533 a(n) = (1/n) * Sum_{k=1..n} floor(n/k) * n^k.

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A344814 a(n) = Sum_{k=1..n} floor(n/k) * 3^(k-1).

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A344815 a(n) = Sum_{k=1..n} floor(n/k) * 4^(k-1).

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A344816 a(n) = Sum_{k=1..n} floor(n/k) * 5^(k-1).

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