A222548 -id:A222548 - cp's OEIS frontend

A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.

1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 10, 1, 33, 83, 74, 32, 14, 1, 65, 245, 274, 136, 52, 16, 1, 129, 731, 1058, 644, 254, 66, 20, 1, 257, 2189, 4162, 3160, 1396, 382, 92, 23, 1, 513, 6563, 16514, 15692, 8054, 2502, 596, 115, 27, 1, 1025, 19685, 65794, 78256, 47452, 17086, 4388, 833, 147, 29

Offset: 1

Seiichi Manyama, May 27 2021

			Square array begins:
   1,  1,   1,    1,    1,     1, ...
   3,  5,   9,   17,   33,    65, ...
   5, 11,  29,   83,  245,   731, ...
   8, 22,  74,  274, 1058,  4162, ...
  10, 32, 136,  644, 3160, 15692, ...
  14, 52, 254, 1396, 8054, 47452, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..5 give A006218, A222548, A318742, A318743, A318744.
T(n,n) gives A332469.
Cf. A306533, A344726.

T[n_, k_] := Sum[Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 27 2021 *)

PARI
```
T(n, k) = sum(j=1, n, (n\j)^k);
```

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

from math import isqrt
from itertools import count, islice
def A344725_T(n,k): return -(s:=isqrt(n))**(k+1)+sum((q:=n//w)*(w**k-(w-1)**k+q**(k-1)) for w in range(1,s+1))
def A344725_gen(): # generator of terms
     return (A344725_T(k+1,n-k) for n in count(1) for k in range(n))
A344725_list = list(islice(A344725_gen(),30)) # Chai Wah Wu, Oct 26 2023

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

T(n,k) = Sum_{j=1..n} Sum_{d|j} d^k - (d - 1)^k.

A344720 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^2.

Original entry on oeis.org

1, 3, 9, 12, 22, 30, 44, 48, 71, 83, 105, 115, 141, 157, 201, 206, 240, 266, 304, 318, 378, 402, 448, 460, 519, 547, 623, 641, 699, 747, 809, 815, 907, 943, 1035, 1064, 1138, 1178, 1286, 1302, 1384, 1448, 1534, 1560, 1710, 1758, 1852, 1866, 1977, 2039, 2179, 2209, 2315, 2395, 2535

Offset: 1

Seiichi Manyama, May 27 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column k=2 of A344726.

Cf. A059851, A078471, A222548.

a[n_] := Sum[(-1)^(k + 1) * Quotient[n, k]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 27 2021 *)
Accumulate[Table[-2*DivisorSigma[0, 2*n] + 3*DivisorSigma[0, n] + 2*DivisorSigma[1, 2*n] - 4*DivisorSigma[1, n], {n, 1, 50}]] (* Vaclav Kotesovec, May 28 2021 *)

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

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

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1+x^k))/(1-x))

a(n) = Sum_{k=1,..n} Sum_{d|k} (-1)^(k/d + 1) * (2*d - 1).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 + x^k).
a(n) ~ Pi^2 * n^2 / 12. - Vaclav Kotesovec, May 28 2021

A350107 a(n) = Sum_{k=1..n} k * floor(n/k)^2.

Original entry on oeis.org

1, 6, 14, 31, 45, 81, 101, 150, 191, 253, 285, 401, 439, 527, 623, 752, 802, 979, 1035, 1233, 1369, 1509, 1577, 1901, 2020, 2186, 2362, 2642, 2728, 3136, 3228, 3549, 3765, 3983, 4215, 4772, 4882, 5126, 5382, 5932, 6054, 6630, 6758, 7202, 7664, 7960, 8100, 8936

Offset: 1

Seiichi Manyama, Dec 14 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column 2 of A350106.

Cf. A000005 (tau), A024916, A143127, A222548, A350123, A350128.

a[n_] := Sum[k * Floor[n/k]^2, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Dec 14 2021 *)
Accumulate[Table[2*k*DivisorSigma[0, k] - DivisorSigma[1, k], {k, 1, 100}]] (* Vaclav Kotesovec, Dec 16 2021 *)

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

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

my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, (2*k-1)*x^k/(1-x^k)^2)/(1-x))

a(n) = sum(k=1, n, 2*k*numdiv(k)-sigma(k));

from math import isqrt
def A350107(n): return -(s:=isqrt(n))**3*(s+1)+sum((q:=n//k)*((k<<1)*((q<<1)+1)-q-1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 24 2023

a(n) = Sum_{k=1..n} k * Sum_{d|k} (2*d - 1)/d = 2 * A143127(n) - A024916(n).
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * x^k/(1 - x^k)^2.
a(n) = Sum_{k=1..n} 2 * k * tau(k) - sigma(k).

A350128 a(n) = Sum_{k=1..n} k^n * floor(n/k)^2.

Original entry on oeis.org

1, 8, 44, 417, 4545, 69905, 1207937, 24904806, 575256641, 14947281595, 427836523971, 13429362462839, 457637290140469, 16843379604615375, 665494379869134005, 28102480944522059434, 1262906802939553227382, 60182948301301262753877

Offset: 1

Seiichi Manyama, Dec 15 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..386

Cf. A222548, A319194, A350107, A350125.

Table[Sum[k^n Floor[n/k]^2,{k,n}],{n,20}] (* Harvey P. Dale, Feb 11 2022 *)

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

a(n) = sum(k=1, n, 2*k*sigma(k, n-1)-sigma(k, n));

from math import isqrt
from sympy import bernoulli
def A350128(n): return (((s:=isqrt(n))+1)*(1-s)*(bernoulli(n+1,s+1)-(b:=bernoulli(n+1)))+sum(k**n*(n+1)*(((q:=n//k)+1)*(q-1))+(1-2*k)*(b-bernoulli(n+1,q+1)) for k in range(1,s+1)))//(n+1) # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{k=1..n} 2 * k * sigma_{n-1}(k) - sigma_{n}(k).

a(n) ~ n^n / (1 - exp(-1)). - Vaclav Kotesovec, Dec 16 2021

A236632 Sum of all divisors of all positive integers <= n minus the total number of divisors of all positive integers <= n.

Original entry on oeis.org

0, 1, 3, 7, 11, 19, 25, 36, 46, 60, 70, 92, 104, 124, 144, 170, 186, 219, 237, 273, 301, 333, 355, 407, 435, 473, 509, 559, 587, 651, 681, 738, 782, 832, 876, 958, 994, 1050, 1102, 1184, 1224, 1312, 1354, 1432, 1504, 1572, 1618, 1732, 1786, 1873, 1941

Offset: 1

Omar E. Pol, Jan 31 2014

			For n = 6 the sets of divisors of the positive integers <= 6 are {1}, {1, 2}, {1, 3}, {1, 2, 4}, {1, 5}, {1, 2, 3, 6}. There are 14 total divisors and their sum is 1 + 3 + 4 + 7 + 6 + 12 = 33, so a(6) = 33 - 14 = 19.

Robert Israel, Table of n, a(n) for n = 1..10000

Partial sums of A065608.

Cf. A000005, A000203, A006218, A024916, A222548.

[(&+[DivisorSigma(1, k) - DivisorSigma(0, k) : k in [1..n]]): n in [1..60]]; // Vincenzo Librandi, Aug 02 2019

A236632:=n->(1/2)*add(floor(n/i)*floor((n-i)/i), i=1..n): seq(A236632(n), n=1..100); # Wesley Ivan Hurt, Jan 30 2016
N:= 1000: # to get a(1) to a(N)
A065608:= Vector(N):
for a from 1 to floor(sqrt(N)) do for b from a to N/a do
   if b = a then
     A065608[a*b] := A065608[a*b] + a - 1
   else
     A065608[a*b] := A065608[a*b] + a + b - 2;
   fi
od od:
ListTools:-PartialSums(convert(A065608,list)); # Robert Israel, May 16 2016

Table[Sum[Floor[n/i]*Floor[(n - i)/i], {i, n}]/2, {n, 50}] (* Wesley Ivan Hurt, Jan 30 2016 *)
Table[Sum[Binomial[Floor[n/i], 2], {i, n}], {n, 51}] (* Michael De Vlieger, May 15 2016 *)
Accumulate@ Table[DivisorSum[n, # - 1 &], {n, 51}] (* or *)
Table[Sum [(k - 1) Floor[n/k], {k, n}], {n, 51}] (* Michael De Vlieger, Apr 03 2017 *)

a(n) = sum(i=1, n, sigma(i)) - sum(i=1, n, numdiv(i)); \\ Michel Marcus, Feb 01 2014

from math import isqrt
def A236632(n): return (s:=isqrt(n))**2*(1-s)+sum((q:=n//k)*((k<<1)+q-3) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 23 2023

a(n) = A024916(n) - A006218(n).
a(n) = (1/2) * Sum_{i=1..n} floor(n/i) * floor((n-i)/i). - Wesley Ivan Hurt, Jan 30 2016
a(n) = Sum_{i=1..n} binomial(floor(n/i),2). - Wesley Ivan Hurt, May 08 2016
a(n) = Sum_{k=1..n} (k-1) * floor(n/k). - Wesley Ivan Hurt, Apr 02 2017
a(n) = (1/2)*(A222548(n) - A006218(n)). - Ridouane Oudra, Aug 01 2019

A332623 a(n) = Sum_{k=1..n} ceiling(n/k)^2.

Original entry on oeis.org

1, 5, 14, 25, 43, 58, 87, 106, 141, 171, 212, 239, 302, 333, 386, 439, 507, 546, 631, 674, 765, 834, 911, 962, 1091, 1157, 1246, 1331, 1450, 1513, 1666, 1733, 1866, 1967, 2080, 2181, 2373, 2452, 2577, 2694, 2883, 2970, 3171, 3262, 3437, 3600, 3749, 3848, 4107, 4225

Offset: 1

Ilya Gutkovskiy, Feb 17 2020

nonn

Cf. A000005, A000203, A006218, A006590, A024916, A222548, A332490, A332569, A332624.

[&+[Ceiling(n/k)^2:k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 17 2020

Table[Sum[Ceiling[n/k]^2, {k, 1, n}], {n, 1, 50}]
Table[n + Sum[2 DivisorSigma[1, k] + DivisorSigma[0, k], {k, 1, n - 1}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[x/(1 - x)^2 + x/(1 - x) Sum[(2 k + 1) x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

from math import isqrt
def A332623(n): return n-(s:=isqrt(n-1))**2*(s+2)+sum((q:=(n-1)//k)*((k<<1)+q+3) for k in range(1,s+1)) # Chai Wah Wu, Oct 24 2023

G.f.: x/(1 - x)^2 + (x/(1 - x)) * Sum_{k>=1} (2*k + 1) * x^k / (1 - x^k).
a(n) = n + Sum_{k=1..n-1} (2*sigma(k) + d(k)).
a(n) ~ n^2 * Pi^2 / 6. - Vaclav Kotesovec, Feb 20 2020

A332569 a(n) = Sum_{k=1..n} floor(n/k) * ceiling(n/k).

Original entry on oeis.org

1, 5, 12, 23, 36, 54, 74, 97, 125, 156, 186, 226, 268, 306, 354, 409, 458, 515, 574, 636, 710, 778, 838, 922, 1013, 1086, 1168, 1264, 1350, 1452, 1556, 1651, 1762, 1864, 1966, 2105, 2234, 2332, 2448, 2594, 2726, 2864, 3004, 3132, 3294, 3444, 3564, 3736, 3917, 4067

Offset: 1

Ilya Gutkovskiy, Feb 16 2020

nonn

Row sums of A085383.

Cf. A000203, A006218, A006590, A024916, A049697, A222548, A332490.

[&+[Floor(n/k)*Ceiling(n/k):k in [1..n]]:n in [1..50]]; // Marius A. Burtea, Feb 16 2020

Table[Sum[Ceiling[n/k] Floor[n/k], {k, 1, n}], {n, 1, 50}]
Table[1 + Sum[DivisorSigma[1, k] + DivisorSigma[1, k + 1], {k, 1, n - 1}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[((1 + x)/(1 - x)) Sum[x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, my(q=n/k); floor(q) * ceil(q)); \\ Michel Marcus, Feb 17 2020

from math import isqrt
from sympy import divisor_sigma
def A332569(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-divisor_sigma(n) # Chai Wah Wu, Oct 23 2023

G.f.: ((1 + x) / (1 - x)) * Sum_{k>=1} x^k / (1 - x^k)^2.
a(n) = 1 + Sum_{k=1..n-1} (sigma(k) + sigma(k+1)) for n > 0.
a(n) ~ (Pi*n)^2/6. - Vaclav Kotesovec, Jun 24 2021

A360610 Triangle read by rows: T(n,k) is the number of squares of side length k that can be placed inside a square of side length n without overlap, 1 <= k <= n.

Original entry on oeis.org

1, 4, 1, 9, 1, 1, 16, 4, 1, 1, 25, 4, 1, 1, 1, 36, 9, 4, 1, 1, 1, 49, 9, 4, 1, 1, 1, 1, 64, 16, 4, 4, 1, 1, 1, 1, 81, 16, 9, 4, 1, 1, 1, 1, 1, 100, 25, 9, 4, 4, 1, 1, 1, 1, 1, 121, 25, 9, 4, 4, 1, 1, 1, 1, 1, 1, 144, 36, 16, 9, 4, 4, 1, 1, 1, 1, 1, 1, 169, 36, 16, 9, 4, 4, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Torlach Rush, Feb 13 2023

T(n,k) is square 1 <= k <= n.
Alternative triangle construction: Write each column k as each square repreated k times.
T(*,1) is A000290.
T(*,2) is A008794.
T(*,3) is A211547.
T(*,4) is A295643(n+4).
T(*,5) is A287392(n+1).
Row sums of triangle are A222548.
This assumes the sides of the small squares are parallel to those of the large square.  If the small squares are allowed to be rotated, better packings may exist (see e.g. the Friedman link).

			Sum_{T(1,*)} = A222548(1) = 1;
Sum_{T(2,*)} = A222548(2) = 5;
Sum_{T(3,*)} = A222548(3) = 11.
Triangle begins:
    1;
    4,  1;
    9,  1, 1;
   16,  4, 1, 1;
   25,  4, 1, 1, 1;
   36,  9, 4, 1, 1, 1;
   49,  9, 4, 1, 1, 1, 1;
   64, 16, 4, 4, 1, 1, 1, 1;
   81, 16, 9, 4, 1, 1, 1, 1, 1;
  100, 25, 9, 4, 4, 1, 1, 1, 1, 1;
  ...

E. Friedman, Packing Unit Squares in Squares: A Survey and New Results, The Electronic Journal of Combinatorics 5 (2009), DS#7.

Cf. A000290, A008794, A211547, A222548, A287392, A295643.

Python
```
def T(n, k): return (n//k)**2
```

T(n,k) = floor(n/k)^2.

cp's OEIS Frontend

A344725 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where T(n,k) = Sum_{j=1..n} floor(n/j)^k.

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A344720 a(n) = Sum_{k=1..n} (-1)^(k+1) * floor(n/k)^2.

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A350107 a(n) = Sum_{k=1..n} k * floor(n/k)^2.

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A350128 a(n) = Sum_{k=1..n} k^n * floor(n/k)^2.

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A236632 Sum of all divisors of all positive integers <= n minus the total number of divisors of all positive integers <= n.

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Magma

Maple

Mathematica

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A332623 a(n) = Sum_{k=1..n} ceiling(n/k)^2.

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Magma

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A332569 a(n) = Sum_{k=1..n} floor(n/k) * ceiling(n/k).

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