A024916 -id:A024916 - cp's OEIS frontend

A318743 a(n) = Sum_{k=1..n} floor(n/k)^4.

1, 17, 83, 274, 644, 1396, 2502, 4388, 6919, 10743, 15385, 22407, 30233, 41209, 53853, 70650, 88636, 113308, 138654, 172332, 207984, 252416, 298002, 358654, 417873, 492065, 569061, 663427, 756053, 875541, 989063, 1130915, 1272967, 1441383, 1607147, 1817080

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

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

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318742, A318744.

[&+[Floor(n/k)^4:k in [1..n]]:n in [1..36]]; // Marius A. Burtea, Jul 16 2019

Table[Sum[Floor[n/k]^4, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[-DivisorSigma[0, k] + 4*DivisorSigma[1, k] - 6*DivisorSigma[2, k] + 4*DivisorSigma[3, k], {k, 1, 40}]]

a(n) = sum(k=1, n, (n\k)^4); \\ Michel Marcus, Sep 03 2018

from math import isqrt
def A318743(n): return -(s:=isqrt(n))**5+sum((q:=n//k)*(k**4-(k-1)**4+q**3) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023

a(n) = -A006218(n) + 4*A024916(n) - 6*A064602(n) + 4*A064603(n).
a(n) ~ zeta(4) * n^4.
a(n) ~ Pi^4 * n^4 / 90.
G.f.: (1/(1 - x)) * Sum_{k>=1} (2*k - 1) * (2*k^2 - 2*k + 1) * x^k/(1 - x^k). - Ilya Gutkovskiy, Jul 16 2019

A318744 a(n) = Sum_{k=1..n} floor(n/k)^5.

Original entry on oeis.org

1, 33, 245, 1058, 3160, 8054, 17086, 33860, 60353, 103437, 164489, 257945, 380407, 556001, 779865, 1085840, 1457122, 1958008, 2544540, 3312306, 4205650, 5336264, 6618976, 8254674, 10059777, 12298021, 14792045, 17829881, 21130663, 25189011, 29518163, 34749419

Offset: 1

Vaclav Kotesovec, Sep 02 2018

nonn

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

Cf. A000005, A000203, A024916, A064602, A064603, A064604.

Cf. A006218, A222548, A318742, A318743.

Table[Sum[Floor[n/k]^5, {k, 1, n}], {n, 1, 40}]
Accumulate[Table[DivisorSigma[0, k] - 5*DivisorSigma[1, k] + 10*DivisorSigma[2, k] - 10*DivisorSigma[3, k] + 5*DivisorSigma[4, k], {k, 1, 40}]]

a(n) = sum(k=1, n, (n\k)^5); \\ Michel Marcus, Sep 03 2018

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (k^5-(k-1)^5)*x^k/(1-x^k))/(1-x)) \\ Seiichi Manyama, May 27 2021

from math import isqrt
def A318744(n): return -(s:=isqrt(n))**6+sum((q:=n//k)*(k**5-(k-1)**5+q**4) for k in range(1,s+1)) # Chai Wah Wu, Oct 26 2023

a(n) = A006218(n) - 5*A024916(n) + 10*A064602(n) - 10*A064603(n) + 5*A064604(n).

a(n) ~ zeta(5) * n^5.

A326616 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), n>=0, A185283(n)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 2, 5, 1, 9, 13, 9, 44, 42, 10, 96, 225, 150, 9, 152, 680, 1098, 576, 3, 155, 1350, 4155, 5201, 2266, 124, 2180, 11730, 26642, 26904, 9966, 140, 3751, 30300, 106281, 182000, 149832, 47466, 160, 6050, 69042, 348061, 896392, 1229760, 855240, 237019

Offset: 0

Alois P. Heinz, Sep 12 2019

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

			T(3,2) = 2: 2a1b, 2b1a.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1,  9,  13;
            9,  44,   42;
           10,  96,  225,   150;
            9, 152,  680,  1098,    576;
            3, 155, 1350,  4155,   5201,   2266;
               124, 2180, 11730,  26642,  26904,   9966;
               140, 3751, 30300, 106281, 182000, 149832, 47466;
               ...

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Partition (number theory)

Main diagonal gives A178682.
Row sums give A326648.
Column sums give A326650.
Cf. A000203, A185283, A326617 (this triangle read by columns), A326649, A326651.

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=h(n)..n), n=0..12);

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++,  If[g[k] >= n, Return[k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t,   b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]][i*j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, h[n], n}], {n, 0, 12}]  // Flatten (* Jean-François Alcover, Feb 27 2021, after Alois P. Heinz *)

Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).

Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).

A332490 a(n) = Sum_{k=1..n} k * ceiling(n/k).

Original entry on oeis.org

1, 4, 10, 18, 30, 42, 61, 77, 101, 124, 153, 177, 218, 246, 285, 325, 373, 409, 467, 507, 570, 624, 683, 731, 816, 873, 942, 1010, 1095, 1155, 1258, 1322, 1418, 1500, 1589, 1673, 1801, 1877, 1976, 2072, 2203, 2287, 2426, 2514, 2643, 2767, 2886, 2982, 3155, 3262

Offset: 1

Ilya Gutkovskiy, Feb 16 2020

nonn

Cf. A000203, A000217, A006590, A024916.

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

Table[Sum[k Ceiling[n/k], {k, 1, n}], {n, 1, 50}]
Table[n (n + 1)/2 + Sum[DivisorSigma[1, k], {k, 1, n - 1}], {n, 1, 50}]
nmax = 50; CoefficientList[Series[x/(1 - x)^3 + (x/(1 - x)) Sum[x^k/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

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

from math import isqrt
def A332490(n): return n*(n+1)-(s:=isqrt(n-1))**2*(s+1)+sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 22 2023

G.f.: x/(1 - x)^3 + (x/(1 - x)) * Sum_{k>=1} x^k / (1 - x^k)^2.
a(n) = n*(n + 1)/2 + Sum_{k=1..n-1} sigma(k).
a(n) ~ (6 + Pi^2)*n^2/12. - Vaclav Kotesovec, Mar 10 2020

A185283 Least k such that sigma(1) + sigma(2) + sigma(3) +...+ sigma(k) >= n.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Michel Lagneau, Jan 21 2012

nonn

			a(3) = 2 because sigma(1) + sigma(2) + sigma(3) = 1+3+4 > 3.

Alois P. Heinz, Table of n, a(n) for n = 0..100000 (terms n = 1..1000 from G. C. Greubel)

Cf. A000203, A024916.

b:= proc(n) option remember; `if`(n=0, 0,
       numtheory[sigma](n)+b(n-1))
    end:
a:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, a(n-1)) do if b(k)>=n then return k fi od
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Sep 12 2019

a[n_] := (k = 1; While[ Total[ DivisorSigma[1, Range[k]]] < n, k++]; k); Table[ a[n], {n, 1, 90}]
Module[{nn=10,ad,th},ad={#[[1]],#[[2]]}&/@Partition[Accumulate[ DivisorSigma[ 1,Range[nn]]],2,1];th=Thread[{Range[2,nn],ad}];Join[ {0,1},Flatten[Table[#[[1]],#[[2,2]]-#[[2,1]]]&/@th]]] (* Harvey P. Dale, Aug 29 2020 *)

a(n) ~ c * sqrt(n), where c = 2*sqrt(3)/Pi. - Amiram Eldar, Dec 27 2024

a(0)=0 prepended by Alois P. Heinz, Sep 12 2019

A191831 a(n) = Sum_{i+j=n, i,j >= 1} tau(i)*sigma(j), where tau() = A000005(), sigma() = A000203().

Original entry on oeis.org

0, 1, 5, 12, 24, 39, 60, 87, 113, 158, 189, 249, 286, 372, 402, 516, 545, 696, 709, 886, 912, 1125, 1110, 1401, 1348, 1674, 1654, 1992, 1906, 2390, 2226, 2735, 2648, 3141, 2926, 3705, 3346, 4069, 3898, 4604, 4223, 5282, 4707, 5757, 5426, 6326, 5754, 7269, 6324, 7669, 7230, 8468, 7556, 9456, 8240, 10018, 9320, 10748, 9621, 12246

Offset: 1

N. J. A. Sloane, Jun 17 2011

This is Andrews's D_{0,1}(n).
From Omar E. Pol, Dec 08 2021: (Start)
Zero together with the convolution of A000005 and A000203.
Zero together with the convolution of A341062 and A024916.
Zero together with the convolution of the nonzero terms of A006218 and A340793.
a(n) is also the volume of a symmetric polycube which belongs to the family of symmetric polycubes that represent the convolution of A000203 with any other integer sequence, n >= 1. (End)

George E. Andrews, Stacked lattice boxes, Ann. Comb. 3 (1999), 115-130.

Cf. A000005, A000203 A006218, A024916, A086718, A237593, A340793, A341062.

with(numtheory); D01:=n->add(tau(j)*sigma(n-j),j=1..n-1);
[seq(D01(n),n=1..60)];

Table[Sum[DivisorSigma[0, j] DivisorSigma[1, n - j], {j, n - 1}], {n, 60}] (* Michael De Vlieger, Jan 01 2017 *)

a(n)=sum(i=1,n-1,numdiv(i)*sigma(n-i)) \\ Charles R Greathouse IV, Feb 19 2013

G.f.: (Sum_{k>=1} x^k/(1 - x^k))*(Sum_{k>=1} k*x^k/(1 - x^k)). - Ilya Gutkovskiy, Jan 01 2017

A226647 Numbers k such that Sum_{i=1..k} sigma(i) is divisible by Sum_{i=1..k} d(i), where sigma(i) = sum of divisors of i (A000203), and d(i) = number of divisors of i (A000005).

Original entry on oeis.org

1, 9, 25, 37, 63, 71876888199

Offset: 1

Alex Ratushnyak, Jun 13 2013

No other terms below 2^36. - Alex Ratushnyak, Jun 29 2013

The ratio corresponding to a(6) is 4249100789716352394810 / 1807894350282 = 2350303705. a(7) > 10^12. - Giovanni Resta, Apr 13 2017

			A006218(9) = 23, A024916(9) = 69, 23 divides 69, so 9 is in the sequence.

C. K. Caldwell, and G. L. Honaker, Jr., Prime curio for 37

Cf. A000005, A000203, A006218, A024916.

Flatten[Position[Accumulate[Table[{DivisorSigma[1,n],DivisorSigma[0,n]},{n,100}]],?(Divisible[First[#],Last[#]]&),{1},Heads->False]] (* _Harvey P. Dale, Jun 19 2013 *)

isok(n) = sum(k=1, n, sigma(k)) % sum(k=1, n, numdiv(k)) == 0; \\ Michel Marcus, Jul 07 2016

Numbers k such that A006218(k) divides A024916(k).

a(6) from Giovanni Resta, Apr 12 2017

A244048 Antisigma(n) minus the sum of remainders of n mod k, for k = 1,2,3,...,n.

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 12, 13, 20, 24, 32, 33, 49, 50, 60, 69, 84, 85, 106, 107, 129, 140, 154, 155, 191, 197, 213, 226, 254, 255, 297, 298, 329, 344, 364, 377, 432, 433, 455, 472, 522, 523, 577, 578, 618, 651, 677, 678, 754, 762, 805, 826

Offset: 1

Omar E. Pol, Jun 23 2014

nonn

For n > 1 a(n) is the sum of all aliquot parts of all positive integers < n. - Omar E. Pol, Mar 27 2021

			From _Omar E. Pol_, Mar 27 2021: (Start)
The following diagrams show a square dissection into regions that are the symmetric representation of A000203, A004125, A153485 and this sequence.
In order to construct every diagram we use the following rules:
At stage 1 in the first quadrant of the square grid we draw the symmetric representation of sigma(n) using the two Dyck paths described in the rows n and n-1 of A237593.
At stage 2 we draw a pair of orthogonal line segments (if it's necessary) such that in the drawing appears totally formed a square n X n. The area of the region that is above the symmetric representation of sigma(n) equals A004125(n).
At stage 3 we draw a zig-zag path with line segments of length 1 from (0,n-1) to (n-1,0) such that appears a staircase with n-1 steps. The area of the region (or regions) that is below the symmetric representation of sigma(n) and above the staircase equals a(n).
At stage 4 we draw a copy of the symmetric representation of A004125(n) rotated 180 degrees such that one of its vertices is the point (0,0). The area of the region (or regions) that is above of this region and below the staircase equals A153485(n).
Illustration for n = 1..6:
.                                                                    _ _ _ _ _ _
.                                                     _ _ _ _ _     |_ _ _  |_ R|
.                                        _ _ _ _ R   |_ _S_|  R|    | |_T | S |_|
.                             _ _ _ R   |_ _  |_|    | |_  |_ _|    |   |_|_ _  |
.                    _ _     |_S_|_|    | |_|_S |    |_U_|_T | |    |_  U |_T | |
.             _ S   |_ S|   U|_|_|S|    |_ U|_| |    |   | |_|S|    | |_    |_| |
.            |_|    |_|_|    |_|_|_|    |_|_ _|_|    |_V_|_U_|_|    |_V_|_ _ _|_|
.                  U        V   U       V
.
n:            1       2         3           4             5               6
R: A004125    0       0         1           1             4               3
S: A000203    1       3         4           7             6              12
T: a(n)       0       0         1           2             5               6
U: A153485    0       1         2           5             6              12
V: A004125    0       0         1           1             4               3
.
Illustration for n = 7..9:
.                                                      _ _ _ _ _ _ _ _ _
.                                _ _ _ _ _ _ _ _      |_ _ _S_ _|       |
.            _ _ _ _ _ _ _      |_ _ _ _  |     |     | |_      |_ _ R  |
.           |_ _S_ _|     |     | |_    | |_ R  |     |   |_    |_ S|   |
.           | |_    |_ R  |     |   |_  |_S |_ _|     |     |_  T |_|_ _|
.           |   |_  T |_ _|     |     |_T |_ _  |     |_ _    |_      | |
.           |_ _  |_    | |     |_ _  U |_    | |     |   |  U  |_    | |
.           |   |_U |_  |S|     |   |_    |_  | |     |   |_ _    |_  |S|
.           |  V  |   |_| |     |  V  |     |_| |     |  V    |     |_| |
.           |_ _ _|_ _ _|_|     |_ _ _|_ _ _ _|_|     |_ _ _ _|_ _ _ _|_|
.
n:                 7                    8                      9
R: A004125         8                    8                     12
S: A000203         8                   15                     12
T: a(n)           12                   13                     20
U: A153485        13                   20                     24
V: A004125         8                    8                     12
.
Illustration for n = 10..12:
.                                                         _ _ _ _ _ _ _ _ _ _ _ _
.                              _ _ _ _ _ _ _ _ _ _ _     |_ _ _ _ _ _  |         |
.     _ _ _ _ _ _ _ _ _ _     |_ _ _S_ _ _|         |    | |_        | |_ _   R  |
.    |_ _ _S_ _  |       |    | |_        |      R  |    |   |_      |     |_    |
.    | |_      | |_  R   |    |   |_      |_        |    |     |_    |_  S   |   |
.    |   |_    |_ _|_    |    |     |_      |_      |    |       |_    |_    |_ _|
.    |     |_      | |_ _|    |       |_   T  |_ _ _|    |         |_ T  |_ _ _  |
.    |       |_ T  |_ _  |    |_ _ _    |_        | |    |_ _        |_        | |
.    |_ _      |_      | |    |     |_ U  |_      | |    |   |    U    |_      | |
.    |   |_ U    |_    |S|    |       |_    |_    |S|    |   |_          |_    | |
.    |     |_      |_  | |    |         |     |_  | |    |     |_ _        |_  | |
.    |  V    |       |_| |    |  V      |       |_| |    |  V      |         |_| |
.    |_ _ _ _|_ _ _ _ _|_|    |_ _ _ _ _|_ _ _ _ _|_|    |_ _ _ _ _|_ _ _ _ _ _|_|
.
n:            10                         11                          12
R: A004125    13                         22                          17
S: A000203    18                         12                          28
T: a(n)       24                         32                          33
U: A153485    32                         33                          49
V: A004125    13                         22                          17
.
Note that in the diagrams the symmetric representation of a(n) is the same as the symmetric representation of A153485(n-1) rotated 180 degrees.
The original examples (dated Jun 24 2014) were only the diagrams for n = 11 and n = 12. (End)

Paolo Xausa, Table of n, a(n) for n = 1..10000

Also zero together with A153485.

Cf. A000203, A000290, A001065, A067439, A024816, A024916, A067439, A123327, A196020, A236104, A237270, A237271, A237593, A244049, A244251, A262626, A342344.

With[{r=Range[100]},Join[{0},Accumulate[DivisorSigma[1,r]-r]]] (* Paolo Xausa, Oct 16 2023 *)

from math import isqrt
def A244048(n): return (-n*(n-1)-(s:=isqrt(n-1))**2*(s+1) + sum((q:=(n-1)//k)*((k<<1)+q+1) for k in range(1,s+1)))>>1 # Chai Wah Wu, Oct 22 2023

a(n) = A024816(n) - A004125(n).
a(n) = A000217(n) - A000203(n) - A004125(n).
a(n) = A024916(n) - A000203(n) - A000217(n-1).
a(n) = A000217(n) - A123327(n).
a(n) = A153485(n-1), n >= 2.

A294016 a(n) = sum of all divisors of all positive integers <= n, minus the sum of remainders of n mod k, for k = 1, 2, 3, ..., n.

Original entry on oeis.org

1, 4, 7, 14, 17, 30, 33, 48, 57, 74, 77, 110, 113, 134, 153, 184, 187, 230, 233, 278, 301, 330, 333, 406, 419, 452, 479, 536, 539, 624, 627, 690, 721, 762, 789, 900, 903, 948, 983, 1084, 1087, 1196, 1199, 1280, 1347, 1400, 1403, 1556, 1573, 1660, 1703, 1796, 1799, 1932, 1967, 2096, 2143, 2208, 2211, 2428, 2431, 2500

Offset: 1

Omar E. Pol, Oct 22 2017

nonn

a(n) is also the area (also the number of cells) of the n-th polygon formed by the Dyck path described in A237593 and its mirror, as shown below in the example.

a(n) is also the volume (and the number of cubes) in the n-th level (starting from the top) of the pyramid described in A294017.

			Illustration of initial terms:
.
.   _ 1
.  |_|_ _ 4
.    |   |
.    |_ _|_ _   7
.        |   |_
.        |_    |
.          |_ _|_ _ _  14
.              |     |_
.              |       |
.              |_      |
.                |_ _ _|_ _ _
.                      |     |  17
.                      |     |_ _
.                      |_ _      |
.                          |     |
.                          |_ _ _|_ _ _ _
.                                |       |_  30
.                                |         |_
.                                |           |
.                                |_          |
.                                  |_        |
.                                    |_ _ _ _|_ _ _ _
.                                            |       |
.                                            |       |_  33
.                                            |         |_ _
.                                            |_ _          |
.                                                |_        |
.                                                  |       |
.                                                  |_ _ _ _|
.

Partial sums of A294015.
Partial sums gives A294017.
Cf. A000203, A000217, A000290, A013661, A004125, A024816, A024916, A067436, A153485, A196020, A235791, A236104, A237591, A237593, A244048, A245092.

A294016 := proc(n)
    A024916(n)-A004125(n) ;
end proc:
seq(A294016(n),n=1..80) ; # R. J. Mathar, Nov 07 2017

Accumulate[Table[2*(DivisorSigma[1, n] - n) + 1, {n, 1, 100}]] (* Amiram Eldar, Mar 30 2024 *)

from math import isqrt
def A294016(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))-n**2 # Chai Wah Wu, Oct 22 2023

a(n) = A024916(n) - A004125(n).
a(n) = A000290(n) - A067436(n).
From Omar E. Pol, Nov 05 2017: (Start)
a(n) = A000203(n) + A024816(n) + A153485(n) - A004125(n).
a(n) = A000217(n) + A153485(n) - A004125(n).
a(n) = A000203(n) + A153485(n) + A244048(n). (End)
a(n) = (Pi^2/6 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 30 2024

A294017 Partial sums of A294016.

Original entry on oeis.org

1, 5, 12, 26, 43, 73, 106, 154, 211, 285, 362, 472, 585, 719, 872, 1056, 1243, 1473, 1706, 1984, 2285, 2615, 2948, 3354, 3773, 4225, 4704, 5240, 5779, 6403, 7030, 7720, 8441, 9203, 9992, 10892, 11795, 12743, 13726, 14810, 15897, 17093, 18292, 19572, 20919, 22319, 23722, 25278, 26851, 28511, 30214, 32010, 33809

Offset: 1

Omar E. Pol, Oct 22 2017

nonn

a(n) is also the volume of another version of the pyramid with n levels (starting from the top) described in A245092. Both pyramids have the same top view and the same front view, but in this pyramid the volume in the n-th level is equal to A294016(n) instead of A024916(n).

Cf. A000203, A072481, A175254, A237593, A245092, A294015, A294016.

from math import isqrt
def A294017(n): return (((s:=isqrt(n))**2*(s+1)*((s+1)*((s<<1)+1)-6*(n+1))>>1) + sum((q:=n//k)*(-k*(q+1)*(3*k+(q<<1)+1)+3*(n+1)*((k<<1)+q+1)) for k in range(1,s+1)))//3-n*(n+1)*((n<<1)+1)//6 # Chai Wah Wu, Oct 22 2023

a(n) = A175254(n) - A072481(n).

cp's OEIS Frontend

A318743 a(n) = Sum_{k=1..n} floor(n/k)^4.

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A318744 a(n) = Sum_{k=1..n} floor(n/k)^5.

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A326616 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), n>=0, A185283(n)<=k<=n, read by rows.

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A332490 a(n) = Sum_{k=1..n} k * ceiling(n/k).

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A185283 Least k such that sigma(1) + sigma(2) + sigma(3) +...+ sigma(k) >= n.

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A191831 a(n) = Sum_{i+j=n, i,j >= 1} tau(i)*sigma(j), where tau() = A000005(), sigma() = A000203().

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A226647 Numbers k such that Sum_{i=1..k} sigma(i) is divisible by Sum_{i=1..k} d(i), where sigma(i) = sum of divisors of i (A000203), and d(i) = number of divisors of i (A000005).

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