A024916 -id:A024916 - cp's OEIS frontend

1, 3, 6, 8, 13, 16, 23, 25, 30, 35, 46, 46, 59, 66, 75, 74, 91, 91, 110, 112, 125, 136, 159, 152, 169, 182, 195, 199, 228, 223, 254, 253, 274, 291, 316, 297, 334, 353, 378, 373, 414, 409, 452, 460, 475, 498, 545, 520, 557, 565, 598, 608, 661, 652, 693, 690

Offset: 1

Hieronymus Fischer, Jul 05 2007, Jul 15 2007, Jan 07 2009

Sums of rows of the triangle in A138530. - Reinhard Zumkeller, Mar 26 2008

			5 = 11111(base 1) = 101(base 2) = 12(base 3) = 11(base 4) = 10(base 5). Thus a(5) = ds_1(5)+ds_2(5)+ds_3(5)+ds_4(5)+ds_5(5) = 5+2+3+2+1 = 13.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Sum

Cf. A131384, A007953.

Table[n + Total@ Map[Total@ IntegerDigits[n, #] &, Range[2, n]], {n, 56}] (* Michael De Vlieger, Jan 03 2017 *)

a(n)=sum(i=2,n+1,vecsum(digits(n,i))); \\ R. J. Cano, Jan 03 2017

a(n) = n^2-sum{k>0, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = n^2-sum{2<=p<=n, (p-1)*sum{0

a(n) = n^2-A024916(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

a(n) = A004125(n)+A006218(n)-sum{k>1, sum{2<=p<=n, (p-1)*floor(n/p^k)}}.

Asymptotic behavior: a(n) = (1-Pi^2/12)*n^2 + O(n*log(n)) = A004125(n) + A006218(n) + O(n*log(n)).

Lim a(n)/n^2 = 1 - Pi^2/12 for n-->oo.

G.f.: (1/(1-x))*(x(1+x)/(1-x)^2-sum{k>0,sum{j>1,(j-1)*x^(j^k)/(1-x^(j^k))}= }).

Also: (1/(1-x))*(x(1+x)/(1-x)^2-sum{m>1, sum{10,j^(1/k) is an integer, j^(1/k)-1}}*x^m}).

a(n) = n^2-sum{10,sum{1

Recurrence: a(n)=a(n-1)-b(n)+2n-1, where b(n)=sum{1

a(n) = sum{1<=p<=n, ds_p(n)} where ds_p = digital sum base p.

a(n) = A043306(n) + n (that sequence ignores unary) = A014837(n) + n + 1 (that sequence ignores unary and base n in which n is "10"). - Alonso del Arte, Mar 26 2009

A244970 Total number of regions after n-th stage in the diagram of the symmetric representation of sigma on the four quadrants.

Original entry on oeis.org

1, 2, 6, 7, 11, 12, 16, 17, 25, 29, 33, 34, 38, 42, 50, 51, 55, 56, 60, 61, 73, 77, 81, 82, 90, 94, 106, 107, 111, 112, 116, 117, 129, 133, 141, 142, 146, 150, 162, 163, 167, 168, 172, 176, 184, 188, 192, 193, 201, 209, 221, 225, 229, 230, 242, 243, 255, 259, 263, 264

Offset: 1

Omar E. Pol, Jul 08 2014

nonn

Partial sums of A244971.
If we use toothpicks of length 1/2, so the area of the central square is equal to 1. The total area of the structure after n-th stage is equal to A024916(n), the sum of all divisors of all positive integers <= n, hence the total area of the n-th set of symmetric regions added at n-th stage is equal to sigma(n) = A000203(n), the sum of divisors of n.
If we use toothpicks of length 1, so the number of cells (and the area) of the central square is equal to 4. The number of cells (and the total area) of the structure after n-th stage is equal to 4*A024916(n) = A243980(n), hence the number of cells (and the total area) of the n-th set of symmetric regions added at n-th stage is equal to 4*A000203(n) = A239050(n).
a(n) is also the total number of terraces of the stepped pyramid with n levels described in A244050. - Omar E. Pol, Apr 20 2016

			Illustration of the structure after 15 stages (contains 50 regions):
.
.                   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                  |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.               _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _
.             _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_
.           _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_
.          |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  |
.     _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _
.    | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | |
.    | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | |
.    | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | |
.    | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | |
.    | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | |
.    | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | |
.    | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | |
.    | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | |
.    | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | |
.    | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | |
.    | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | |
.    | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | |
.    | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | |
.    | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | |
.    | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | |
.    |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_|
.          | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |
.          |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _|
.            |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|
.              |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|
.                  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.                  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
The diagram is also the top view of the stepped pyramid with 15 levels described in A244050. - _Omar E. Pol_, Apr 20 2016

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237590, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244370, A244371, A244971, A245092.

A248076 Partial sums of the sum of the 5th powers of the divisors of n: Sum_{i=1..n} sigma_5(i).

Original entry on oeis.org

1, 34, 278, 1335, 4461, 12513, 29321, 63146, 122439, 225597, 386649, 644557, 1015851, 1570515, 2333259, 3415660, 4835518, 6792187, 9268287, 12572469, 16673621, 21988337, 28424681, 36677981, 46446732, 58699434, 73107634, 90873690, 111384840, 136555392

Offset: 1

Wesley Ivan Hurt, Sep 30 2014

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

Cf. A001160 (sigma_5).
Cf. A024916: Partial sums of sigma(n)   = A000203(n).
Cf. A064602: Partial sums of sigma_2(n) = A001157(n).
Cf. A064603: Partial sums of sigma_3(n) = A001158(n).
Cf. A064604: Partial sums of sigma_4(n) = A001159(n).

[(&+[DivisorSigma(5,j): j in [1..n]]): n in [1..30]]; // G. C. Greubel, Nov 07 2018

with(numtheory): A248076:=n->add(sigma[5](i), i=1..n): seq(A248076(n), n=1..50);

Table[Sum[DivisorSigma[5, i], {i, n}], {n, 30}]
Accumulate[DivisorSigma[5, Range[30]]] (* Vaclav Kotesovec, Mar 30 2018 *)

lista(nn) = vector(nn, n, sum(i=1, n, sigma(i, 5))) \\ Michel Marcus, Sep 30 2014

from math import isqrt
def A248076(n): return ((s:=isqrt(n))**3*(s+1)**2*(1-2*s*(s+1)) + sum((q:=n//k)*(12*k**5+q*(q**2*(q*(2*q+6)+5)-1)) for k in range(1,s+1)))//12 # Chai Wah Wu, Oct 21 2023

a(n) = Sum_{i=1..n} sigma_5(i) = Sum_{i=1..n} A001160(i).
a(n) ~ Zeta(6) * n^6 / 6. - Vaclav Kotesovec, Sep 02 2018
a(n) ~ Pi^6 * n^6 / 5670. - Vaclav Kotesovec, Sep 02 2018
a(n) = Sum_{k=1..n} (Bernoulli(6, floor(1 + n/k)) - 1/42)/6, where Bernoulli(n,x) are the Bernoulli polynomials. - Daniel Suteu, Nov 07 2018
a(n) = Sum_{k=1..n} k^5 * floor(n/k). - Daniel Suteu, Nov 08 2018

A256533 Product of n and the sum of all divisors of all positive integers <= n.

Original entry on oeis.org

1, 8, 24, 60, 105, 198, 287, 448, 621, 870, 1089, 1524, 1833, 2310, 2835, 3520, 4046, 4986, 5643, 6780, 7791, 8954, 9913, 11784, 13050, 14664, 16308, 18480, 20010, 22860, 24614, 27424, 29865, 32606, 35245, 39528, 42032, 45448, 48828, 53680, 56744, 62160, 65532, 70752, 75870, 80868, 84882, 92640, 97363, 104000

Offset: 1

Omar E. Pol, May 02 2015

nonn

a(n) is also sum of the volumes (or the total number of unit cubes) from two complementary polycubes: the irregular staircase after n-th stage described in A244580, and the irregular stepped pyramid after (n-1)st stage described in A245092. Note that in both structures the horizontal area in the n-th level is also the symmetric representation of sigma(n). This comment is represented by the third formula.

			For n = 3; a(3) = 3 * 8 = 19 + 5 = 24.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A000203, A000578, A024916, A064987, A143128, A175254, A196020, A236104, A237270, A237271, A237593, A244580, A245092, A245100, A256532, A332264.

a[n_]:=n*Apply[Plus,Flatten[Divisors[Range[n]]]]; Array[a,50] (* Ivan N. Ianakiev, May 03 2015 *)
nxt[{n_,sd_,a_}]:=Module[{k=(n+1)*(DivisorSigma[1,n+1]+sd)},{n+1,sd+DivisorSigma[ 1,n+1],k}]; NestList[ nxt,{1,1,1},50][[;;,3]] (* Harvey P. Dale, Jun 12 2023 *)

a(n) = n*sum(k=1, n, n\k*k); \\ Michel Marcus, Apr 29 2020

def A256533(n):
    s=0
    for k in range(1, n+1):
        s+=n%k
    return (n**3)-(s*n) # Indranil Ghosh, Feb 13 2017

from math import isqrt
def A256533(n): return n*(-(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) = n*A024916(n).
a(n) = n^3 - A256532(n).
a(n) = A143128(n) + A175254(n-1), n > 1.
a(n) = A332264(n) + A175254(n). - Omar E. Pol, Apr 29 2020

A271342 Sum of all even divisors of all positive integers <= n.

Original entry on oeis.org

0, 2, 2, 8, 8, 16, 16, 30, 30, 42, 42, 66, 66, 82, 82, 112, 112, 138, 138, 174, 174, 198, 198, 254, 254, 282, 282, 330, 330, 378, 378, 440, 440, 476, 476, 554, 554, 594, 594, 678, 678, 742, 742, 814, 814, 862, 862, 982, 982, 1044, 1044, 1128, 1128, 1208, 1208, 1320, 1320, 1380, 1380, 1524, 1524, 1588, 1588, 1714, 1714

Offset: 1

Omar E. Pol, Apr 08 2016

a(n) is also the sum of all even divisors of all even positive integers <= n.
a(n) is also the total number of parts in all partitions of all positive integers <= n into an even number of equal parts. - Omar E. Pol, Jun 04 2017
The bisection of this sequence equals twice A024916 (see formulas). - Michel Marcus, Dec 14 2017

			For n = 6 the divisors of all positive integers <= 6 are [1], [1, 2], [1, 3], [1, 2, 4], [1, 5], [1, 2, 3, 6] and the even divisors of all positive integers <= 6 are [2], [2, 4], [2, 6], so a(6) = 2 + 2 + 4 + 2 + 6 = 16. On the other hand the sum of all the divisors of all positive integers <= 6/2 are [1] + [1 + 2] + [1 + 3] = A024916(3) = 8, so a(6) = 2*8 = 16.
For n = 10, (floor(10/2) = 5) numbers have divisor 2, (floor(10/4) = 2) numbers have divisor 4, ..., (floor(10/10) = 1) numbers have divisor 10. Therefore, a(10) = 5 * 2 + 2 * 4 + 1 * 6 + 1 * 8 + 1 * 10 = 42. - _David A. Corneth_, Jun 06 2017

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000203, A006128, A024916, A078471, A146076, A222171, A271343.

Partial sums of A146076.

Accumulate@ Array[DivisorSum[#, # &, EvenQ] &, 65] (* Michael De Vlieger, Jun 06 2017 *)

a(n) = sum(k=1, n, sumdiv(k, d, (1-d%2)*d)); \\ Michel Marcus, Jun 05 2017

a(n) = 2 * sum(k=1, n\2, k*(n\(k<<1))) \\ David A. Corneth, Jun 06 2017

def A271342(n): return sum(k*((n>>1)//k) for k in range(1, (n>>1)+1))<<1 # Chai Wah Wu, Apr 26 2023

from math import isqrt
def A271342(n): return -(s:=isqrt(m:=n>>1))**2*(s+1) + sum((q:=m//k)*((k<<1)+q+1) for k in range(1,s+1)) # Chai Wah Wu, Oct 21 2023

a(1) = 0.
a(n) = 2*A024916((n-1)/2), if n is odd and n > 1.
a(n) = 2*A024916(n/2), if n is even.
a(n) = A024916(n) - A078471(n).
For n > 1, a(2*n + 1) = a(2*n). - David A. Corneth, Jun 06 2017
a(n) = c * n^2 + O(n*log(n)), where c = Pi^2/24 = 0.411233... (A222171). - Amiram Eldar, Nov 27 2023

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14

Offset: 1

Omar E. Pol, Oct 02 2016

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Indranil Ghosh, Rows 1..100 of triangle, flattened
Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

A275670 G.f. A(x,y) satisfies: A(x,y) = xy + A(x,xy)^2, with A(0,y) = 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 4, 0, 8, 1, 0, 16, 4, 0, 32, 14, 0, 64, 40, 0, 128, 108, 2, 0, 256, 272, 12, 0, 512, 664, 52, 0, 1024, 1568, 188, 0, 2048, 3632, 608, 1, 0, 4096, 8256, 1816, 12, 0, 8192, 18528, 5128, 76, 0, 16384, 41088, 13856, 360, 0, 32768, 90304, 36176, 1446, 0, 65536, 196864, 91856, 5192, 4, 0, 131072, 426368, 227968, 17192, 42, 0, 262144, 918016, 555040, 53504, 284, 0, 524288, 1966848, 1329696, 158588, 1496, 0, 1048576, 4195328, 3141632, 451824, 6704, 0, 2097152, 8914432, 7334208, 1245936, 26772, 6

Offset: 0

Paul D. Hanna, Aug 04 2016

Compare g.f. to G(x,y) = x*y + G(x*y,y)^2 with G(0,y) = 0, which generates triangle A138157.
Apparently, the g.f. of column n equals y^n*x^A033156(n) * P(n,x)/Q(n,x), where:
Q(n,x) = Product_{k=1..n} (1 - 2*x^k)^floor(n/k),
and P(n,x) is of degree A024916(n) - A033156(n).

			G.f.: A(x,y) = 1 + y*x + 2*y*x^2 + 4*y*x^3 + (y^2 + 8*y)*x^4 + (4*y^2 + 16*y)*x^5 + (14*y^2 + 32*y)*x^6 + (40*y^2 + 64*y)*x^7 + (2*y^3 + 108*y^2 + 128*y)*x^8 + (12*y^3 + 272*y^2 + 256*y)*x^9 + (52*y^3 + 664*y^2 + 512*y)*x^10 + (188*y^3 + 1568*y^2 + 1024*y)*x^11 + (y^4 + 608*y^3 + 3632*y^2 + 2048*y)*x^12 +...
such that A(x,y) = x*y + A(x,x*y)^2, with A(0,y) = 1; further,
A(x,y) = x*y + ( x^2*y + A(x,x^2*y)^2 )^2,
A(x,y) = x*y + ( x^2*y + ( x^3*y + A(x,x^3*y)^2 )^2 )^2, etc.
This table of coefficients in g.f. A(x,y) begins:
1;
0, 1;
0, 2;
0, 4;
0, 8, 1;
0, 16, 4;
0, 32, 14;
0, 64, 40;
0, 128, 108, 2;
0, 256, 272, 12;
0, 512, 664, 52;
0, 1024, 1568, 188;
0, 2048, 3632, 608, 1;
0, 4096, 8256, 1816, 12;
0, 8192, 18528, 5128, 76;
0, 16384, 41088, 13856, 360;
0, 32768, 90304, 36176, 1446;
0, 65536, 196864, 91856, 5192, 4;
0, 131072, 426368, 227968, 17192, 42;
0, 262144, 918016, 555040, 53504, 284;
0, 524288, 1966848, 1329696, 158588, 1496;
0, 1048576, 4195328, 3141632, 451824, 6704;
0, 2097152, 8914432, 7334208, 1245936, 26772, 6;
0, 4194304, 18876416, 16943680, 3342784, 98060, 80;
0, 8388608, 39848960, 38785536, 8761720, 335704, 636;
0, 16777216, 83890176, 88063616, 22508448, 1088496, 3844;
0, 33554432, 176166912, 198506624, 56822624, 3375096, 19492;
0, 67108864, 369106944, 444562432, 141270272, 10080760, 87184, 4;
0, 134217728, 771764224, 989807872, 346507120, 29167000, 354628, 80;
0, 268435456, 1610629120, 2192154880, 839762496, 82113648, 1338376, 812;
0, 536870912, 3355467776, 4831741952, 2013427136, 225746384, 4753320, 5916;
0, 1073741824, 6979354624, 10603063808, 4781027584, 607828752, 16052296, 35000;
0, 2147483648, 14495563776, 23174734336, 11254280416, 1606760304, 51954808, 178904, 1; ...
Row polynomials begin:
n=0: 1;
n=1: y;
n=2: 2*y;
n=3: 4*y;
n=4: 8*y + y^2;
n=5: 16*y + 4*y^2;
n=6: 32*y + 14*y^2;
n=7: 64*y + 40*y^2;
n=8: 128*y + 108*y^2 + 2*y^3;
n=9: 256*y + 272*y^2 + 12*y^3;
n=10: 512*y + 664*y^2 + 52*y^3;
n=11: 1024*y + 1568*y^2 + 188*y^3;
n=12: 2048*y + 3632*y^2 + 608*y^3 + y^4;
n=13: 4096*y + 8256*y^2 + 1816*y^3 + 12*y^4;
n=14: 8192*y + 18528*y^2 + 5128*y^3 + 76*y^4;
n=15: 16384*y + 41088*y^2 + 13856*y^3 + 360*y^4;
n=16: 32768*y + 90304*y^2 + 36176*y^3 + 1446*y^4;
n=17: 65536*y + 196864*y^2 + 91856*y^3 + 5192*y^4 + 4*y^5; ...
the first row in which y^m appears is given by n = A033156(m), where A033156 begins:
[1, 4, 8, 12, 17, 22, 27, 32, 38, 44, 50, 56, 62, 68, 74, 80, 87, 94, 101, 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 200, ...].
Generating functions of initial columns.
G.f. of column 0: 1
G.f. of column 1: y*x/(1-2*x).
G.f. of column 2: y^2*x^4/((1-2*x)^2*(1-2*x^2)).
G.f. of column 3: y^3*2*x^8/((1-2*x)^3*(1-2*x^2)*(1-2*x^3)).
G.f. of column 4: y^4*x^12*(1 + 4*x - 10*x^3)/((1-2*x)^4*(1-2*x^2)^2*(1-2*x^3)*(1-2*x^4)).
G.f. of column 5: y^5*x^17*(4 + 2*x + 8*x^2 - 28*x^4)/((1-2*x)^5*(1-2*x^2)^2*(1-2*x^3)*(1-2*x^4)*(1-2*x^5)).
G.f. of column 6: y^6*x^22*(6 + 8*x - 20*x^3 - 24*x^4 - 36*x^5 - 56*x^6 + 16*x^7 + 176*x^8 + 224*x^9 - 336*x^11)/((1-2*x)^6*(1-2*x^2)^3*(1-2*x^3)^2*(1-2*x^4)*(1-2*x^5)*(1-2*x^6)).
G.f. of column 7: y^7*x^27*(4 + 24*x + 4*x^2 - 12*x^3 - 72*x^5 - 112*x^6 - 96*x^7 + 112*x^8 - 64*x^9 + 64*x^10 + 496*x^11 + 576*x^12 - 1056*x^14) / ((1-2*x)^7*(1-2*x^2)^3*(1-2*x^3)^2*(1-2*x^4)*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)).
G.f. of column 8: y^8*x^32*(1 + 24*x + 36*x^2 - 4*x^3 - 88*x^4 - 202*x^5 - 14*x^6 - 82*x^7 - 168*x^8 + 400*x^9 + 440*x^10 + 892*x^11 + 1292*x^12 - 660*x^13 - 800*x^14 - 688*x^15 - 1776*x^16 - 1136*x^17 - 4504*x^18 - 2672*x^19 + 4672*x^20 + 5664*x^21 + 12672*x^22 - 13728*x^24) / ((1-2*x)^8*(1-2*x^2)^4*(1-2*x^3)^2*(1-2*x^4)^2*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)).
G.f. of column 9: y^9*x^38*(8 + 60*x + 72*x^2 + 16*x^3 - 238*x^4 - 584*x^5 - 232*x^6 + 172*x^7 + 328*x^8 + 52*x^9 + 1012*x^10 + 2636*x^11 + 1464*x^12 + 520*x^13 - 2040*x^14 - 664*x^15 - 2360*x^16 - 8712*x^17 - 13008*x^18 - 3696*x^19 + 12080*x^20 + 15392*x^21 + 1456*x^22 - 11040*x^23 + 18112*x^24 + 37728*x^25 + 47040*x^26 - 34304*x^27 - 78144*x^28 - 73216*x^29 + 91520*x^31) / ((1-2*x)^9*(1-2*x^2)^4*(1-2*x^3)^3*(1-2*x^4)^2*(1-2*x^5)*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)*(1-2*x^9)).
G.f. of column 10: y^10*x^44*(28 + 96*x + 198*x^2 - 160*x^3 - 864*x^4 - 596*x^5 - 856*x^6 - 384*x^7 + 3652*x^8 + 4752*x^9 + 696*x^10 - 2972*x^11 + 3928*x^12 + 4848*x^13 - 8360*x^14 - 18768*x^15 - 11000*x^16 - 14184*x^17 - 9896*x^18 + 17184*x^19 + 23664*x^20 + 7904*x^21 + 34480*x^22 + 53472*x^23 + 54160*x^24 + 68160*x^25 + 10560*x^26 - 166208*x^27 - 203488*x^28 - 86720*x^29 - 23552*x^30 + 13632*x^31 + 67584*x^32 - 95232*x^33 - 232256*x^34 + 129536*x^35 + 677632*x^36 + 624000*x^37 + 355840*x^38 - 67584*x^39 - 988416*x^40 - 1464320*x^41 + 1244672*x^43) / ((1-2*x)^10*(1-2*x^2)^5*(1-2*x^3)^3*(1-2*x^4)^2*(1-2*x^5)^2*(1-2*x^6)*(1-2*x^7)*(1-2*x^8)*(1-2*x^9)*(1-2*x^10)).
...
The g.f. of column n, y^n * x^A033156(n) * P(n,x)/Q(n,x), appears to have the following denominator:
Q(n,x) = Product_{k=1..n} (1 - 2*x^k)^floor(n/k), with
P(n,x) being a polynomial of degree A024916(n) - A033156(n),
where A024916(n) = Sum_{k=1..n} k*floor(n/k).
...

Paul D. Hanna, Table of n, a(n) for n = 0..1606, for rows 0..122 of triangle in flattened form.

Cf. A274965 (row sums), A275691 (antidiagonal sums), A033156.

Cf. variant: A138157.

/* Print first N rows of this triangle: */ N=32;
{a(n) = my(A=1 +x*O(x^n)); for(k=0, n, A = A^2 + y*x^(n+1-k)); polcoeff(A, n)}
{for(n=0, N, for(k=0,n, if(k==0,print1(polcoeff(a(n)+y*O(y^n),k,y)", "), if(polcoeff(a(n)+y*O(y^n),k,y)==0,break,print1(polcoeff(a(n)+y*O(y^n),k,y),", "))));print(""))}

G.f. A(x,y) satisfies: 1 = ...(((((A(x,y) - x*y)^(1/2) - x^2*y)^(1/2) - x^3*y)^(1/2) - x^4*y)^(1/2) - x^5*y)^(1/2) -...- x^n*y)^(1/2) -..., an infinite series of nested square roots.

A299692 a(n) is the total area that is visible in the perspective view of the stepped pyramid with n levels described in A245092.

Original entry on oeis.org

3, 10, 20, 35, 51, 75, 97, 128, 159, 197, 231, 283, 323, 375, 429, 492, 544, 619, 677, 759, 833, 913, 983, 1091, 1172, 1266, 1360, 1472, 1560, 1692, 1786, 1913, 2027, 2149, 2267, 2430, 2542, 2678, 2812, 2982, 3106, 3286, 3416, 3588, 3756, 3920, 4062, 4282, 4437, 4630, 4804, 5006, 5166, 5394, 5576, 5808, 6002

Offset: 1

Omar E. Pol, Mar 06 2018

nonn

a(n) is also the sum of all divisors of all positive integers <= n, plus the n-th oblong number, since A024916(n) equals the total area of the horizontal terraces of the stepped pyramid with n levels, and A002378(n) equals the total area of the vertical sides that are visible (see link).

a(n) is also the sum of all aliquot divisors of all positive integers <= n, plus the n-th triangular matchstick number.

			For n = 3 the areas of the terraces of the first three levels starting from the top of the stepped pyramid are 1, 3 and 4 respectively. On the other hand the areas of the vertical sides that are visible are [1, 1], [2, 2], [2, 1, 1, 2], or in successive levels 2, 4, 6 respectively. Hence the total area that is visible is equal to 1 + 3 + 4 + 2 + 4 + 6 = 8 + 12 = 20, so a(3) = 20.
For n = 16 the total number of horizontal and vertical cells that are visible are 220 and 272 respectively. So a(16) = 220 + 272 = 492 (see the link).

Omar E. Pol, Perspective view of the pyramid with 16 levels, contains 492 cells.

Partial sums of A224880.

Cf. A002378, A024916, A045943, A072691, A153485, A196020, A236104, A237048, A237270, A237271, A237591, A237593, A244050, A245092, A262626, A328366.

Accumulate[Table[DivisorSigma[1, n] + 2*n, {n, 1, 50}]] (* Amiram Eldar, Mar 21 2024 *)

a(n) = sum(k=1, n, n\k*k) + n*(n+1); \\ Michel Marcus, Jun 21 2018

from math import isqrt
def A299692(n): return n*(n+1)+(-(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) = A024916(n) + A002378(n).
a(n) = A153485(n) + A045943(n).
a(n) = A328366(n)/2. - Omar E. Pol, Apr 22 2020
a(n) = c * n^2 + O(n*log(n)), where c = zeta(2)/2 + 1 = A072691 + 1 = 1.822467... . - Amiram Eldar, Mar 21 2024

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A112964 Sum(mu(i)*sigma(j): i+j=n), with mu=A008683 and sigma=A000203.

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A127096 Triangle T(n,m) = A000012*A127094 read by rows.

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A131383 Total digital sum of n: sum of the digital sums of n for all the bases 1 to n (a 'digital sumorial').

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A244970 Total number of regions after n-th stage in the diagram of the symmetric representation of sigma on the four quadrants.

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A248076 Partial sums of the sum of the 5th powers of the divisors of n: Sum_{i=1..n} sigma_5(i).

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A256533 Product of n and the sum of all divisors of all positive integers <= n.

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A271342 Sum of all even divisors of all positive integers <= n.

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A275670 G.f. A(x,y) satisfies: A(x,y) = xy + A(x,xy)^2, with A(0,y) = 1.