A024916 -id:A024916 - cp's OEIS frontend

A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...

1, 2, 3, 8, 9, 10, 11, 16, 26, 27, 28, 33, 34, 35, 36, 57, 58, 68, 69, 74, 75, 76, 77, 82, 108, 109, 119, 124, 125, 126, 127, 148, 149, 150, 151, 201, 202, 203, 204, 209, 210, 211, 212, 217, 227, 228, 229, 250, 300, 326, 327, 332, 333, 343, 344, 349, 350, 351, 352, 357, 358, 359, 369, 454, 455, 456

Offset: 1

Ilya Gutkovskiy, Jul 13 2019

nonn

Partial sums of A035316.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of a(n)/n^(3/2) for n = 1..10000

Cf. A013936, A024916, A035316, A309126, A309127.

Table[Sum[k^2 Floor[n/k^2], {k, 1, n}], {n, 1, 66}]
nmax = 66; CoefficientList[Series[1/(1 - x) Sum[k^2 x^(k^2)/(1 - x^(k^2)), {k, 1, Floor[nmax^(1/2)] + 1}], {x, 0, nmax}], x] // Rest

a(n) = sum(k=1, n, k^2*(n\k^2)); \\ Seiichi Manyama, Aug 30 2021

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

a(n) ~ zeta(3/2)*n^(3/2)/3 - n/2. - Vaclav Kotesovec, Aug 30 2021

A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 6, 8, 8, 1, 10, 16, 15, 10, 1, 18, 38, 37, 21, 14, 1, 34, 100, 111, 63, 33, 16, 1, 66, 278, 373, 237, 113, 41, 20, 1, 130, 796, 1335, 999, 489, 163, 56, 23, 1, 258, 2318, 4957, 4461, 2393, 833, 248, 69, 27, 1, 514, 6820, 18831, 20583, 12513, 4795, 1418, 339, 87, 29

Offset: 1

Ilya Gutkovskiy, Dec 09 2018

			Square array begins:
   1,   1,    1,    1,     1,      1,  ...
   3,   4,    6,   10,    18,     34,  ...
   5,   8,   16,   38,   100,    278,  ...
   8,  15,   37,  111,   373,   1335,  ...
  10,  21,   63,  237,   999,   4461,  ...
  14,  33,  113,  489,  2393,  12513,  ...

Index entries for sequences related to sigma(n)

Columns k=0..5 give A006218, A024916, A064602, A064603, A064604, A248076.

Cf. A082771, A109974, A319194 (diagonal).

Table[Function[k, Sum[j^k Floor[n/j] , {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[1/(1 - x) Sum[j^k x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, Sum[DivisorSigma[k, j], {j, 1, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten

from itertools import count, islice
from math import isqrt
from sympy import bernoulli
def A319649_T(n,k): return (((s:=isqrt(n))+1)*(bernoulli(k+1)-bernoulli(k+1,s+1))+sum(w**k*(k+1)*((q:=n//w)+1)-bernoulli(k+1)+bernoulli(k+1,q+1) for w in range(1,s+1)))//(k+1) + int(k==0)
def A319649_gen(): # generator of terms
     return (A319649_T(k+1,n-k-1) for n in count(1) for k in range(n))
A319649_list = list(islice(A319649_gen(),30)) # Chai Wah Wu, Oct 24 2023

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

A(n,k) = Sum_{j=1..n} sigma_k(j).

A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

Original entry on oeis.org

1, 4, 7, 1, 11, 3, 1, 13, 4, 3, 1, 18, 7, 4, 3, 1, 20, 6, 7, 4, 3, 1, 23, 12, 6, 7, 4, 3, 1, 28, 8, 12, 6, 7, 4, 3, 1, 31, 15, 8, 12, 6, 7, 4, 3, 1, 30, 13, 15, 8, 12, 6, 7, 4, 3, 1, 40, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 42, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1, 38, 28, 12, 18, 13, 15, 8, 12, 6, 7, 4, 3, 1

Offset: 1

Omar E. Pol, Jan 12 2021

T(n,k) is the total area (or number of cells) of the terraces that are in the k-th level that contains terraces starting from the base of the symmetric tower (a polycube) described in A221529 which has A000041(n-1) levels in total. The terraces of the polycube are the symmetric representation of sigma. The terraces are in the levels that are the partition numbers A000041 starting from the base. Note that for n >= 2 there are n - 1 terraces because the first terrace of the tower is formed by two symmetric representations of sigma in the same level. The volume (or the number of cubes) equals A066186(n), the sum of all parts of all partitions of n. The volume is also the sum of all divisors of all terms of the first n rows of A336811. That is due to the correspondence between divisors and partitions (cf. A336811). The growth of the volume (A066186) represents the convolution of A000203 and A000041.

			Triangle begins:
   1;
   4;
   7,  1;
  11,  3,  1;
  13,  4,  3,  1;
  18,  7,  4,  3,  1;
  20,  6,  7,  4,  3,  1;
  23, 12,  6,  7,  4,  3,  1;
  28,  8, 12,  6,  7,  4,  3,  1;
  31, 15,  8, 12,  6,  7,  4,  3,  1;
  30, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  40, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  42, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
  38, 28, 12, 18, 13, 15,  8, 12,  6,  7,  4,  3,  1;
...
For n = 7, sigma(7) = 1 + 7 = 8 and sigma(6) = 1 + 2 + 3 + 6 = 12, and 8 + 12 = 20, so the first term of row 7 is T(7,1) = 20. The other terms in row 7 are the first five terms of A000203 in reverse order, that is [6, 7, 4, 3, 1] so the 7th row of the triangle is [20, 6, 7, 4, 3, 1].
From _Omar E. Pol_, Jul 11 2021: (Start)
For n = 7 we can see below the top view and the lateral view of the pyramid described in A245092 (with seven levels) and the top view and the lateral view of the tower described in A221529 (with 11 levels).
                                           _
                                          | |
                                          | |
                                          | |
        _                                 |_|_
       |_|_                               |   |
       |_ _|_                             |_ _|_
       |_ _|_|_                           |   | |
       |_ _ _| |_                         |_ _|_|_
       |_ _ _|_ _|_                       |_ _ _| |_
       |_ _ _ _| | |_                     |_ _ _|_ _|_ _
       |_ _ _ _|_|_ _|                    |_ _ _ _|_|_ _|
.
         Figure 1.                           Figure 2.
        Lateral view                       Lateral view
       of the pyramid.                     of the tower.
.
.       _ _ _ _ _ _ _                      _ _ _ _ _ _ _
       |_| | | | | | |                    |_| | | | |   |
       |_ _|_| | | | |                    |_ _|_| | |   |
       |_ _|  _|_| | |                    |_ _|  _|_|   |
       |_ _ _|    _|_|                    |_ _ _|    _ _|
       |_ _ _|  _|                        |_ _ _|  _|
       |_ _ _ _|                          |       |
       |_ _ _ _|                          |_ _ _ _|
.
          Figure 3.                          Figure 4.
          Top view                           Top view
       of the pyramid.                     of the tower.
.
Both polycubes have the same base which has an area equal to A024916(7) = 41 equaling the sum of the 7th row of triangle.
Note that in the top view of the tower the symmetric representation of sigma(6) and the symmetric representation of sigma(7) appear unified in the level 1 of the structure as shown above in the figure 4 (that is due to the first two partition numbers A000041 are [1, 1]), so T(7,1) = sigma(7) + sigma(6) = 8 + 12 = 20. (End)

The length of row n is A028310(n-1).
Row sums give A024916.
Column 1 gives 1 together with A092403.
Other columns give A000203.
Cf. A175254 (volume of the pyramid).
Cf. A066186 (volume of the tower).
Cf. A346533 (mirror).
Cf. A000041, A221529, A237270, A237593, A245092, A272172 (similar), A336811, A338156, A339106, A340035.

Table[If[n <= 2, {Total@ #}, Prepend[#2, Total@ #1] & @@ TakeDrop[#, 2]] &@ DivisorSigma[1, Range[n, 1, -1]], {n, 14}] // Flatten (* Michael De Vlieger, Jan 13 2021 *)

A345023 a(n) is the surface area of the symmetric tower described in A221529 which is a polycube whose successive terraces are the symmetric representation of sigma A000203(i) (from i = 1 to n) starting from the top and the levels of these terraces are the partition numbers A000041(h-1) (from h = 1 to n) starting from the base.

Original entry on oeis.org

6, 16, 32, 58, 90, 142, 202, 292, 406, 562, 754, 1034, 1370, 1822, 2410, 3176, 4136, 5402, 6982, 9026, 11598, 14838, 18894, 24034, 30396, 38312, 48136, 60288, 75220, 93624, 116104, 143598, 177090, 217770, 267106, 326820, 398804, 485472, 589644, 714564, 864000, 1042524, 1255308

Offset: 1

Omar E. Pol, Jun 05 2021

nonn

The largest side of the base of the tower has length n.
The base of the tower is the symmetric representation of A024916(n).
The volume of the tower is equal to A066186(n).
The area of each lateral view of the tower is equal to A000070(n-1).
The growth of the volume of the tower represents the convolution of A000203 and A000041.
The above results are because the correspondence between divisors and partitions described in A338156 and A336812.
The tower is also a member of the family of the stepped pyramid described in A245092.
The equivalent sequence for the surface area of the stepped pyramid is A328366.

			For n = 7 we can see below some views of two associated polycubes called "prism of partitions" and "tower". Both objects contains the same number of cubes (that property is also valid for n >= 1).
     _ _ _ _ _ _ _
    |_ _ _ _      |                 7
    |_ _ _ _|_    |           4     3
    |_ _ _    |   |             5   2
    |_ _ _|_ _|_  |         3   2   2                                    _
    |_ _ _      | |               6 1                 1                 | |
    |_ _ _|_    | |         3     3 1                 1                 | |
    |_ _    |   | |           4   2 1                 1                 | |
    |_ _|_ _|_  | |       2   2   2 1                 1                _|_|
    |_ _ _    | | |             5 1 1               1 1               |   |
    |_ _ _|_  | | |         3   2 1 1               1 1              _|_ _|
    |_ _    | | | |           4 1 1 1             1 1 1             | |   |
    |_ _|_  | | | |       2   2 1 1 1             1 1 1            _|_|_ _|
    |_ _  | | | | |         3 1 1 1 1           1 1 1 1          _| |_ _ _|
    |_  | | | | | |       2 1 1 1 1 1         1 1 1 1 1      _ _|_ _|_ _ _|
    |_|_|_|_|_|_|_|     1 1 1 1 1 1 1     1 1 1 1 1 1 1     |_ _|_|_ _ _ _|
.
       Figure 1.           Figure 2.        Figure 3.           Figure 4.
   Front view of the      Partitions        Position          Lateral view
  prism of partitions.       of 7.         of the 1's.        of the tower.
.
.
                                                             _ _ _ _ _ _ _
                                                            |   | | | | |_|  1
                                                            |   | | |_|_ _|  2
                                                            |   |_|_  |_ _|  3
                                                            |_ _    |_ _ _|  4
                                                                |_  |_ _ _|  5
                                                                  |       |  6
                                                                  |_ _ _ _|  7
.
                                                               Figure 5.
                                                               Top view
                                                             of the tower.
.
Figure 1 is a two-dimensional diagram of the partitions of 7. The area of the diagram is A066186(7) = 105. Note that the diagram can be interpreted also as the front view of a right prism whose volumen is 1*7*A000041(7) = 1*7*15 = 105, equaling the volume of the tower that appears in the figures 4 and 5.
Figure 2 shows the partitions of 7 in accordance with the diagram.
Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions, see the figures 3 and 4. In this case the mentioned area equals A000070(7-1) = 30.
The connection between these two objects is a representation of the correspondence divisor/part described in A338156. See also A336812.

Cf. A000041, A000070, A000203, A024916, A066186, A176206, A196020, A221529, A236104, A237593, A244050, A245092, A327329, A328366, A336811, A336812, A338156.

Accumulate @ Table[4 * PartitionsP[k-1] + 2 * DivisorSigma[1, k], {k, 1, 50}] (* Amiram Eldar, Jul 14 2021 *)

a(n) = 4*A000070(n-1) + 2*A024916(n).

a(n) = 4*A000070(n-1) + A327329(n).

A364970 a(n) = Sum_{k=1..n} binomial(floor(n/k)+2,3).

Original entry on oeis.org

1, 5, 12, 26, 42, 73, 102, 152, 204, 278, 345, 464, 556, 693, 835, 1021, 1175, 1422, 1613, 1907, 2173, 2496, 2773, 3228, 3569, 4015, 4445, 4998, 5434, 6120, 6617, 7331, 7965, 8717, 9391, 10392, 11096, 12031, 12909, 14059, 14921, 16219, 17166, 18489, 19711, 21072, 22201

Offset: 1

Seiichi Manyama, Oct 23 2023

Partial sums of A007437.

Cf. A006218, A024916, A064602, A365409, A365439.

Table[Sum[Binomial[Floor[n/k+2],3],{k,n}],{n,50}] (* Harvey P. Dale, Aug 04 2024 *)

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

from math import isqrt
def A364970(n): return (-(s:=isqrt(n))**2*(s+1)*(s+2)+sum((q:=n//k)*(3*k*(k+1)+(q+1)*(q+2)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 26 2023

a(n) = Sum_{k=1..n} binomial(k+1,2) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^3 = 1/(1-x) * Sum_{k>=1} binomial(k+1,2) * x^k/(1-x^k).
a(n) = (A064602(n)+A024916(n))/2. - Chai Wah Wu, Oct 26 2023

A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

Original entry on oeis.org

1, 7, 23, 64, 135, 282, 493, 864, 1375, 2166, 3168, 4715, 6536, 9132, 12278, 16525, 21371, 27998, 35314, 44995, 55847, 69504, 84455, 103882, 124428, 150005, 177921, 212017, 247978, 292890, 339267, 395874, 455796, 526692, 600788, 691066, 782457, 891048, 1004814

Offset: 1

Seiichi Manyama, Oct 23 2023

Partial sums of A073570.

Cf. A006218, A024916, A064602, A064603, A064604, A364970, A365409.

a(n) = sum(k=1, n, binomial(n\k+4, 5));

from math import isqrt, comb
def A365439(n): return (-(s:=isqrt(n))**2*comb(s+4,4)+sum((q:=n//k)*(5*comb(k+3,4)+comb(q+4,4)) for k in range(1,s+1)))//5 # Chai Wah Wu, Oct 26 2023

a(n) = Sum_{k=1..n} binomial(k+3,4) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^5 = 1/(1-x) * Sum_{k>=1} binomial(k+3,4) * x^k/(1-x^k).
a(n) = (A064604(n)+6*A064603(n)+11*A064602(n)+6*A024916(n))/24. - Chai Wah Wu, Oct 26 2023

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

Original entry on oeis.org

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

A373882 Number of lattice points inside or on the 4-dimensional hypersphere x^2 + y^2 + z^2 + u^2 = 10^n.

Original entry on oeis.org

9, 569, 49689, 4937225, 493490641, 49348095737, 4934805110729, 493480252693889, 49348022079085897, 4934802199975704129, 493480220066583590433, 49348022005552308828457, 4934802200546833521392241, 493480220054489318828539601, 49348022005446802425711456713, 4934802200544679211736756034457

Offset: 0

Seiichi Manyama, Jun 21 2024

nonn

Wikipedia, Jacobi four square theorem

Cf. A068785, A373881, A373883, A373884, A373885.

Cf. A024916, A046895.

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(4, 10^n);

from math import isqrt
def A373882(n): return 1+((-(s:=isqrt(a:=10**n))**2*(s+1)+sum((q:=a//k)*((k<<1)+q+1) for k in range(1,s+1))&-1)<<2)+(((t:=isqrt(m:=a>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # Chai Wah Wu, Jun 21 2024

a(n) = A046895(10^n).
a(n) == 1 (mod 8).
Limit_{n->oo} a(n) = Pi^2*100^n/2. - Hugo Pfoertner, Jun 21 2024

A067435 a(n) is the sum of all the remainders when n-th odd number is divided by odd numbers < 2n-1.

Original entry on oeis.org

0, 0, 2, 3, 6, 9, 16, 13, 27, 31, 34, 43, 57, 56, 75, 80, 96, 99, 121, 122, 155, 164, 163, 184, 220, 218, 255, 252, 277, 304, 339, 328, 372, 389, 412, 433, 491, 478, 515, 536, 570, 609, 638, 647, 722, 713, 746, 767, 858, 842, 910, 939, 942, 993, 1060, 1057

Offset: 1

Amarnath Murthy, Jan 29 2002

			a(7) = 16 = 1 +3 +6 +4 +2 = 13 % 3 + 13 % 5 + 13 % 7 + 13 % 9 + 13 % 11.

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

Cf. A000203, A000593, A004125, A067436, A024916, A327329.

L:= [seq(4*n-3 - numtheory:-sigma(2*n-1)-numtheory:-sigma((n-1)/2^padic:-ordp(n-1,2)), n=1..100)]:
ListTools:-PartialSums(L); # Robert Israel, Jan 16 2019

from math import isqrt
def A327329(n): return -(s:=isqrt(n))**2*(s+1)+sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))
def A067435(n): return n*((n<<1)-1)-(A327329(n<<1)>>1)-A327329(n>>1)+3*(A327329(n)>>1)+A327329(n-1>>1)-(A327329(n-1)>>1) # Chai Wah Wu, Nov 01 2023

a(n) = a(n-1) + 4*n-3 - A000203(2*n-1) - A000593(n-1). - Robert Israel, Jan 16 2019

a(n) = n*(2*n-1) - A326123(n) - A078471(n-1) = n*(2*n-1) - A024916(2*n) - 2*A024916(floor(n/2)) + 3*A024916(n) + 2*A024916(floor((n-1)/2)) - A024916(n-1). - Chai Wah Wu, Nov 01 2023

Corrected and extended by several contributors.

A076664 a(n) = Sum_{k=1..n} antisigma(k), where antisigma(i) = sum of the nondivisors of i that are between 1 and i.

Original entry on oeis.org

0, 0, 2, 5, 14, 23, 43, 64, 96, 133, 187, 237, 314, 395, 491, 596, 731, 863, 1033, 1201, 1400, 1617, 1869, 2109, 2403, 2712, 3050, 3400, 3805, 4198, 4662, 5127, 5640, 6181, 6763, 7338, 8003, 8684, 9408, 10138, 10957, 11764, 12666, 13572, 14529, 15538

Offset: 1

Joseph L. Pe, Oct 24 2002

Sum of all proper nondivisors of all positive integers <= n. - Omar E. Pol, Feb 11 2014

			a(5) = antisigma(1) + ... + antisigma(5) = 0 + 0 + 2 + 3 + 9 = 14.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A024816, A000292, A024916.

l = {}; s = 0; Do[s = s + (n (n + 1) / 2) - DivisorSigma[1, n]; l = Append[l, s], {n, 1, 100}]; l
Accumulate[Table[Total[Complement[Range[n],Divisors[n]]],{n,50}]] (* Harvey P. Dale, May 19 2014 *)

a(n) = sum(k=1, n, k*(k+1)/2-sigma(k)); \\ Michel Marcus, Sep 18 2017

from math import isqrt
def A076664(n): return n*(n+1)*(n+2)//3+(s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1 # Chai Wah Wu, Oct 22 2023

a(n) = A000292(n) - A024916(n), n >= 1. Omar E. Pol, Feb 11 2014
a(n) = Sum_{k=1..n} Sum_{i=1..k-1} (n-k-i+1) mod (n-i+1). - Wesley Ivan Hurt, Sep 13 2017
G.f.: x/(1 - x)^4 - (1/(1 - x))*Sum_{k>=1} k*x^k/(1 - x^k). - Ilya Gutkovskiy, Sep 18 2017

cp's OEIS Frontend

A309125 a(n) = n + 2^2 * floor(n/2^2) + 3^2 * floor(n/3^2) + 4^2 * floor(n/4^2) + ...

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A319649 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{j=1..n} j^k * floor(n/j).

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A340584 Irregular triangle read by rows T(n,k) in which row n lists sigma(n) + sigma(n-1) together with the first n - 2 terms of A000203 in reverse order, with T(1,1) = 1, n >= 1.

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A364970 a(n) = Sum_{k=1..n} binomial(floor(n/k)+2,3).

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A365439 a(n) = Sum_{k=1..n} binomial(floor(n/k)+4,5).

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

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Python

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A373882 Number of lattice points inside or on the 4-dimensional hypersphere x^2 + y^2 + z^2 + u^2 = 10^n.

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Python

Formula

A067435 a(n) is the sum of all the remainders when n-th odd number is divided by odd numbers < 2n-1.