A239050 -id:A239050 - cp's OEIS frontend

A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

8, 24, 48, 80, 112, 160, 200, 264, 328, 408, 464, 560, 624, 728, 832, 960, 1040, 1184, 1272, 1432, 1576, 1728, 1832, 2024, 2160, 2336, 2512, 2736

Offset: 1

Omar E. Pol, Jun 26 2014

Partial sums of A244371.
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).

			Illustration of the structure after 16 stages (Contains 960 toothpicks):
.
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.

Cf. A000203, A004125, A024916, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A243980, A244050, A244360-A244363, A244371, A244970, A244971, A245092.

a(n) = 4*A244362(n) = 8*A244360(n).

a(8) corrected and more terms from Omar E. Pol, Oct 18 2014

A239052 Sum of divisors of 4*n-2.

Original entry on oeis.org

3, 12, 18, 24, 39, 36, 42, 72, 54, 60, 96, 72, 93, 120, 90, 96, 144, 144, 114, 168, 126, 132, 234, 144, 171, 216, 162, 216, 240, 180, 186, 312, 252, 204, 288, 216, 222, 372, 288, 240, 363, 252, 324, 360, 270, 336, 384, 360, 294, 468, 306, 312, 576

Offset: 1

Omar E. Pol, Mar 09 2014

Bisection of A062731 (odd part).
a(n) is also the total number of cells in the n-th branch of the second quadrant of the spiral formed by the parts of the symmetric representation of sigma(4n-2). For the quadrants 1, 3, 4 see A112610, A239053, A193553. The spiral has been obtained according to the following way: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A237270, see example.
We can find the spiral on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

			Illustration of initial terms:
------------------------------------------------------
.        Branches of the spiral
.        in the second quadrant             n    a(n)
------------------------------------------------------
.
.                  _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|         4     24
.                 | |
.             12 _| |
.               |_ _|  _ _ _ _ _ _
.         12 _ _|     |  _ _ _ _ _|         3     18
.      _ _ _| |    9 _| |
.     |  _ _ _|  9 _|_ _|
.     | |      _ _| |      _ _ _ _
.     | |     |  _ _| 12 _|  _ _ _|         2     12
.     | |     | |      _|   |
.     | |     | |     |  _ _|
.     | |     | |     | |    3 _ _
.     | |     | |     | |     |  _|         1      3
.     |_|     |_|     |_|     |_|
.
For n = 4 the sum of divisors of 4*n-2 is 1 + 2 + 7 + 14 = A000203(14) = 24. On the other hand the parts of the symmetric representation of sigma(14) are [12, 12] and the sum of them is 12 + 12 = 24, equaling the sum of divisors of 14, so a(4) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A008438, A016825, A062731, A074400, A112610, A193553, A196020, A236104, A235791, A237270, A237271, A237591, A237593, A239050, A239053, A244050, A245092, A262626.

a[n_] := DivisorSigma[1, 4*n - 2]; Array[a, 100] (* Amiram Eldar, Dec 17 2022 *)

a(n) = A000203(4n-2) = A000203(A016825(n-1)).
a(n) = 3*A008438(n-1). - Joerg Arndt, Mar 09 2014
Sum_{k=1..n} a(k) = (3*Pi^2/8) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A239053 Sum of divisors of 4*n-1.

Original entry on oeis.org

4, 8, 12, 24, 20, 24, 40, 32, 48, 56, 44, 48, 72, 72, 60, 104, 68, 72, 124, 80, 84, 120, 112, 120, 156, 104, 108, 152, 144, 144, 168, 128, 132, 240, 140, 168, 228, 152, 192, 216, 164, 168, 260, 248, 180, 248, 216, 192, 336, 200, 240, 312, 212, 264, 296

Offset: 1

Omar E. Pol, Mar 09 2014

Bisection of A008438.
a(n) is also the total number of cells in the n-th branch of the third quadrant of the spiral formed by the parts of the symmetric representation of sigma(4n-1), see example. For the quadrants 1, 2, 4 see A112610, A239052, A193553. The spiral has been obtained according to the following way: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> A237270.
We can find the spiral (mentioned above) on the terraces of the pyramid described in A244050. - Omar E. Pol, Dec 06 2016

			Illustration of initial terms:
-----------------------------------------------------
.        Branches of the spiral
.        in the third quadrant             n    a(n)
-----------------------------------------------------
.     _       _       _       _
.    | |     | |     | |     | |
.    | |     | |     | |     |_|_ _
.    | |     | |     | |    2  |_ _|       1      4
.    | |     | |     |_|_     2
.    | |     | |    4    |_
.    | |     |_|_ _        |_ _ _ _
.    | |    6      |_      |_ _ _ _|       2      8
.    |_|_ _ _        |_   4
.   8      | |_ _      |
.          |_    |     |_ _ _ _ _ _
.            |_  |_    |_ _ _ _ _ _|       3     12
.           8  |_ _|  6
.                  |
.                  |_ _ _ _ _ _ _ _
.                  |_ _ _ _ _ _ _ _|       4     24
.                 8
.
For n = 4 the sum of divisors of 4*n-1 is 1 + 3 + 5 + 15 = A000203(15) = 24. On the other hand the parts of the symmetric representation of sigma(15) are [8, 8, 8] and the sum of them is 8 + 8 + 8 = 24, equaling the sum of divisors of 15, so a(4) = 24.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A004767, A008438, A062731, A074400, A112610, A193553, A196020, A235791, A236104, A237270, A237591, A237593, A239050, A239052, A244050, A245092, A262626.

[SumOfDivisors(4*n-1): n in [1..60]]; // Vincenzo Librandi, Dec 07 2016

A239053:=n->numtheory[sigma](4*n-1): seq(A239053(n), n=1..80); # Wesley Ivan Hurt, Dec 06 2016

DivisorSigma[1,4*Range[60]-1] (* Harvey P. Dale, Dec 06 2016 *)
Table[DivisorSigma[1, 4 n - 1], {n, 100}] (* Vincenzo Librandi, Dec 07 2016 *)

a(n) = sigma(4*n-1); \\ Michel Marcus, Dec 07 2016

a(n) = A000203(4n-1) = A000203(A004767(n-1)).
a(n) = 4*A097723(n-1). - Joerg Arndt, Mar 09 2014
Sum_{k=1..n} a(k) = (Pi^2/4) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 17 2022

A244971 Number of regions in the symmetric representation of sigma(n) on the four quadrants.

Original entry on oeis.org

1, 1, 4, 1, 4, 1, 4, 1, 8, 4, 4, 1, 4, 4, 8, 1, 4, 1, 4, 1, 12, 4, 4, 1, 8, 4, 12, 1, 4, 1, 4, 1, 12, 4, 8, 1, 4, 4, 12, 1, 4, 1, 4, 4, 8, 4, 4, 1, 8, 8, 12, 4, 4, 1, 12, 1, 12, 4, 4, 1, 4, 4, 16, 1, 12, 1, 4, 4, 12, 8, 4, 1, 4, 4, 12, 4, 8, 4, 4, 1, 16, 4, 4, 1, 12, 4, 12, 1, 4, 1

Offset: 1

Omar E. Pol, Jul 08 2014

nonn

Partial sums give A244970.

Number of terraces at the n-th level (starting from the top) of the stepped pyramid described in A244050. - Omar E. Pol, Apr 20 2016

			From _Omar E. Pol_, Apr 20 2016: (Start)
Illustration of the top view of the stepped pyramid with 16 levels:
.                 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.                |  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.             _ _| |  _ _ _ _ _ _ _ _ _ _ _ _ _ _  | |_ _
.           _|  _ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _  |_
.         _|  _|  _| |  _ _ _ _ _ _ _ _ _ _ _ _  | |_  |_  |_
.        |  _|   |_ _| |_ _ _ _ _ _ _ _ _ _ _ _| |_ _|   |_  |
.   _ _ _| |  _ _|     |  _ _ _ _ _ _ _ _ _ _  |     |_ _  | |_ _ _
.  |  _ _ _|_| |      _| |_ _ _ _ _ _ _ _ _ _| |_      | |_|_ _ _  |
.  | | |  _ _ _|    _|_ _|  _ _ _ _ _ _ _ _  |_ _|_    |_ _ _  | | |
.  | | | | |  _ _ _| |  _| |_ _ _ _ _ _ _ _| |_  | |_ _ _  | | | | |
.  | | | | | | |  _ _|_|  _|  _ _ _ _ _ _  |_  |_|_ _  | | | | | | |
.  | | | | | | | | |  _ _|   |_ _ _ _ _ _|   |_ _  | | | | | | | | |
.  | | | | | | | | | | |  _ _|  _ _ _ _  |_ _  | | | | | | | | | | |
.  | | | | | | | | | | | | |  _|_ _ _ _|_  | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | |  _ _  | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |   | | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | | | |_ _| | | | | | | | | | | | | | | |
.  | | | | | | | | | | | | | |_|_ _ _ _|_| | | | | | | | | | | | | |
.  | | | | | | | | | | | |_|_  |_ _ _ _|  _|_| | | | | | | | | | | |
.  | | | | | | | | | |_|_    |_ _ _ _ _ _|    _|_| | | | | | | | | |
.  | | | | | | | |_|_ _  |_  |_ _ _ _ _ _|  _|  _ _|_| | | | | | | |
.  | | | | | |_|_ _  | |_  |_ _ _ _ _ _ _ _|  _| |  _ _|_| | | | | |
.  | | | |_|_ _    |_|_ _| |_ _ _ _ _ _ _ _| |_ _|_|    _ _|_| | | |
.  | |_|_ _ _  |     |_  |_ _ _ _ _ _ _ _ _ _|  _|     |  _ _ _|_| |
.  |_ _ _  | |_|_      | |_ _ _ _ _ _ _ _ _ _| |      _|_| |  _ _ _|
.        | |_    |_ _  |_ _ _ _ _ _ _ _ _ _ _ _|  _ _|    _| |
.        |_  |_  |_  | |_ _ _ _ _ _ _ _ _ _ _ _| |  _|  _|  _|
.          |_  |_ _| |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |_ _|  _|
.            |_ _  | |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |  _ _|
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                | |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| |
.                |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
.
Note that the above diagram contains a hidden pattern, simpler, which emerges from the front view of every corner of the stepped pyramid.
For more information about the hidden pattern see A237593.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..5000 (computed from the b-file of A237271 provided by Michel Marcus)

Cf. A000203, A196020, A235791, A236104, A237270, A237271, A237591, A237593, A239050, A239660, A239931-A239934, A244050, A244370, A244371, A244970, A245092.

lista() = {v = readvec("b237271.txt"); for (i=1, #v, vi = v[i]; if (vi == 1, w = 1, w = 4*(vi-1)); print1(w, ", "););} \\ Michel Marcus, Sep 29 2014

a(n) = 1, if A237271(n) = 1.

a(n) = 4*(A237271(n) - 1), if A237271(n) > 1.

A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-1), 1 <= k <= n.

Original entry on oeis.org

1, 3, 0, 4, 0, 1, 7, 0, 3, 1, 6, 0, 4, 3, 2, 12, 0, 7, 4, 6, 2, 8, 0, 6, 7, 8, 6, 4, 15, 0, 12, 6, 14, 8, 12, 4, 13, 0, 8, 12, 12, 14, 16, 12, 7, 18, 0, 15, 8, 24, 12, 28, 16, 21, 8, 12, 0, 13, 15, 16, 24, 14, 28, 28, 24, 12, 28, 0, 18, 13, 30, 16, 48, 24, 49, 32, 36, 14, 14, 0, 12

Offset: 1

Omar E. Pol, Jan 07 2021

Conjecture: the sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.

			Triangle begins:
   1;
   3,  0;
   4,  0,  1;
   7,  0,  3,  1;
   6,  0,  4,  3,  2;
  12,  0,  7,  4,  6,  2;
   8,  0,  6,  7,  8,  6,  4;
  15,  0, 12,  6, 14,  8, 12,  4;
  13,  0,  8, 12, 12, 14, 16, 12,  7;
  18,  0, 15,  8, 24, 12, 28, 16, 21,  8;
  12,  0, 13, 15, 16, 24, 14, 28, 28, 24, 12;
  28,  0, 18, 13, 30, 16, 48, 24, 49, 32, 36, 14;
...
For n = 6 the calculation of every term of row 6 is as follows:
--------------------------
k   A002865         T(6,k)
--------------------------
1      1   *   12  =  12
2      0   *   6   =   0
3      1   *   7   =   7
4      1   *   4   =   4
5      2   *   3   =   6
6      2   *   1   =   2
.           A000203
--------------------------
The sum of row 6 is 12 + 0 + 7 + 4 + 6 + 2 = 31, equaling A138879(6) = 31.

Columns 1, 3 and 4 give A000203.
Column 2 gives A000004.
Columns 5 and 6 gives A074400.
Column 7 and 8 give A239050.
Column 9 gives A319527.
Column 10 gives A319528.
Leading diagonal gives A002865.
Cf. A135010, A138879.

A215947 Difference between the sum of the even divisors and the sum of the odd divisors of 2n.

Original entry on oeis.org

1, 5, 4, 13, 6, 20, 8, 29, 13, 30, 12, 52, 14, 40, 24, 61, 18, 65, 20, 78, 32, 60, 24, 116, 31, 70, 40, 104, 30, 120, 32, 125, 48, 90, 48, 169, 38, 100, 56, 174, 42, 160, 44, 156, 78, 120, 48, 244, 57, 155, 72, 182, 54, 200, 72, 232, 80, 150, 60, 312, 62, 160

Offset: 1

Michel Lagneau, Aug 28 2012

Multiplicative because a(n) = -A002129(2*n), A002129 is multiplicative and a(1) = -A002129(2) = 1. - Andrew Howroyd, Jul 31 2018

			a(6) = 20 because the divisors of 2*6 = 12 are {1, 2, 3, 4, 6, 12} and (12 + 6 + 4 +2) - (3 + 1) = 20.

Michel Lagneau, Table of n, a(n) for n = 1..10000
Hartosh Singh Bal and Gaurav Bhatnagar, Two curious congruences for the sum of divisors function, arXiv:2102.10804 [math.NT], 2021. Mentions this sequence.
H. Movasati and Y. Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2017.

Cf. A000593, A002129, A022998 (Moebius transform), A074400, A195382, A195690.

Cf. A002513, A062731, A111003, A239050.

with(numtheory):for n from 1 to 100 do:x:=divisors(2*n):n1:=nops(x):s0:=0:s1:=0:for m from 1 to n1 do: if irem(x[m],2)=0 then s0:=s0+x[m]:else s1:=s1+x[m]:fi:od:if s0>s1  then printf(`%d, `,s0-s1):else fi:od:

a[n_] := DivisorSum[2n, (1 - 2 Mod[#, 2]) #&];
Array[a, 62] (* Jean-François Alcover, Sep 13 2018 *)
edod[n_]:=Module[{d=Divisors[2n]},Total[Select[d,EvenQ]]-Total[ Select[ d,OddQ]]]; Array[edod,70] (* Harvey P. Dale, Jul 30 2021 *)

a(n) = 4*sigma(n) - sigma(2*n); \\ Andrew Howroyd, Jul 28 2018

From Andrew Howroyd, Jul 28 2018: (Start)
a(n) = 4*sigma(n) - sigma(2*n).
a(n) = -A002129(2*n). (End)
G.f.: Sum_{k>=1} x^k*(1 + 4*x^k + x^(2*k))/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Sep 14 2019
a(p) = p + 1 for p prime >= 3. - Bernard Schott, Sep 14 2019
a(n) = A239050(n) - A062731(n) - Omar E. Pol, Mar 06 2021 (after Andrew Howroyd)
From Amiram Eldar, Nov 18 2022: (Start)
Multiplicative with a(2^e) = 2^(e+2) - 3, and a(p^e) = sigma(p^e) = (p^(e+1) - 1)/(p-1) for p > 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). (End)
Dirichlet g.f.: zeta(s)*zeta(s-1)*(1+2^(1-s)). - Amiram Eldar, Jan 05 2023
From Peter Bala, Sep 25 2023: (Start)
a(2*n) = sigma(2*n) + 2*sigma(n); a(2*n+1) = sigma(2*n+1) = A008438(n)
G.f.: A(q) = Sum_{n >= 1} n*q^n*(1 + 3*q^n)/(1 - q^(2*n)).
Logarithmic g.f.: Sum_{n >= 1} a(n)*q^n/n = Sum_{n >= 1} log(1/(1 - q^n)) + Sum_{n >= 1} log(1/(1 - q^(2*n))) = log (G(q)), where G(q) is the g.f. of A002513. (End)

A239662 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers A017113 interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

Original entry on oeis.org

4, 12, 20, 4, 28, 0, 36, 12, 44, 0, 4, 52, 20, 0, 60, 0, 0, 68, 28, 12, 76, 0, 0, 4, 84, 36, 0, 0, 92, 0, 20, 0, 100, 44, 0, 0, 108, 0, 0, 12, 116, 52, 28, 0, 4, 124, 0, 0, 0, 0, 132, 60, 0, 0, 0, 140, 0, 36, 20, 0, 148, 68, 0, 0, 0, 156, 0, 0, 0, 12, 164, 76, 44, 0, 0, 4, 172, 0, 0, 28, 0, 0, 180, 84, 0, 0, 0, 0, 188, 0, 52, 0, 0, 0

Offset: 1

Omar E. Pol, Mar 30 2014

Gives an identity for A239050. Alternating sum of row n equals A239050(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = 4*A000203(n) = 2*A074400(n) = A239050(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Note that if T(n,k) = 12 then T(n+1,k+1) = 4, the first element of the column k+1.
The number of positive terms in row n is A001227(n).
For more information see A196020.
Column 1 is A017113. - Omar E. Pol, Apr 17 2016

			Triangle begins:
  4;
  12;
  20,   4;
  28,   0;
  36,  12;
  44,   0,  4;
  52,  20,  0;
  60,   0,  0;
  68,  28, 12;
  76,   0,  0,  4;
  84,  36,  0,  0;
  92,   0, 20,  0;
  100, 44,  0,  0;
  108,  0,  0, 12;
  116, 52, 28,  0,  4;
  124,  0,  0,  0,  0;
  132, 60,  0,  0,  0;
  140,  0, 36, 20,  0;
  148, 68,  0,  0,  0;
  156,  0,  0,  0, 12;
  164, 76, 44,  0,  0,  4;
  172,  0,  0, 28,  0,  0;
  180, 84,  0,  0,  0,  0;
  188,  0, 52,  0,  0,  0;
  ...
For n = 9, the 9th row of triangle is [68, 28, 12], therefore the alternating row sum is 68 - 28 + 12 = 52. On the other hand we have that 4*A000203(9) = 2*A074400(9) = A239050(9) = 4*13 = 2*26 = 52, equaling the alternating sum of the 9th row of the triangle.

Cf. A000203, A000217, A003056, A017113, A074400, A196020, A211343, A235791, A236104, A236106, A236112, A237048, A237591, A239050, A239446.

T(n,k) = 2*A236106(n,k) = 4*A196020(n,k).

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.

A246911 Numbers n such that sigma(n+sigma(n)) = 4*sigma(n).

Original entry on oeis.org

28, 66, 348, 496, 840, 920, 1320, 1416, 1602, 1770, 1896, 1920, 2040, 2280, 2556, 3000, 3360, 3720, 4440, 4920, 5456, 5640, 5826, 7080, 7392, 8010, 8040, 8128, 8298, 10528, 10680, 11424, 12768, 12840, 13080, 15108, 15504, 17880, 18120, 18720, 18840, 20832

Offset: 1

Jaroslav Krizek, Sep 07 2014

nonn

			Number 28 (with sigma(28) = 56) is in sequence because sigma(26+sigma(26)) = sigma(84) = 224 = 4*56.

Cf. A000203, A239050, A246456, A246909, A246910, A246912, A246913, A246914.

[n:n in[1..10000] | SumOfDivisors(n+SumOfDivisors(n)) eq 4*SumOfDivisors(n)]

with(numtheory): A246911:=n->`if`(sigma(n+sigma(n)) = 4*sigma(n),n,NULL): seq(A246911(n), n=1..3*10^4); # Wesley Ivan Hurt, Sep 07 2014

for(n=1,10^4,if(sigma(n+sigma(n))==4*sigma(n),print1(n,", "))) \\ Derek Orr, Sep 07 2014

A272026 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers A016945 interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

Original entry on oeis.org

3, 9, 15, 3, 21, 0, 27, 9, 33, 0, 3, 39, 15, 0, 45, 0, 0, 51, 21, 9, 57, 0, 0, 3, 63, 27, 0, 0, 69, 0, 15, 0, 75, 33, 0, 0, 81, 0, 0, 9, 87, 39, 21, 0, 3, 93, 0, 0, 0, 0, 99, 45, 0, 0, 0, 105, 0, 27, 15, 0, 111, 51, 0, 0, 0, 117, 0, 0, 0, 9, 123, 57, 33, 0, 0, 3, 129, 0, 0, 21, 0, 0, 135, 63, 0, 0, 0, 0, 141, 0, 39, 0, 0, 0

Offset: 1

Omar E. Pol, Apr 18 2016

Alternating sum of row n equals 3 times sigma(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = 3*A000203(n) = A272027(n).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
The number of positive terms in row n is A001227(n).
If T(n,k) = 9 then T(n+1,k+1) = 3 is the first element of the column k+1.
For more information see A196020.

			Triangle begins:
    3;
    9;
   15,  3;
   21,  0;
   27,  9;
   33,  0,  3;
   39, 15,  0;
   45,  0,  0;
   51, 21,  9;
   57,  0,  0,  3;
   63, 27,  0,  0;
   69,  0, 15,  0;
   75, 33,  0,  0;
   81,  0,  0,  9;
   87, 39, 21,  0,  3;
   93,  0,  0,  0,  0;
   99, 45,  0,  0,  0;
  105,  0, 27, 15,  0;
  111, 51,  0,  0,  0;
  117,  0,  0,  0,  9;
  123, 57, 33,  0,  0,  3;
  129,  0,  0, 21,  0,  0;
  135, 63,  0,  0,  0,  0;
  141,  0, 39,  0,  0,  0;
  ...
For n = 9 the divisors of 9 are 1, 3, 9, therefore the sum of the divisors of 9 is 1 + 3 + 9 = 13 and 3*13 = 39. On the other hand the 9th row of triangle is 51, 21, 9, therefore the alternating row sum is 51 - 21 + 9 = 39, equaling 3 times sigma(9).

Column 1 is A016945.

Cf. A000203, A001227, A003056, A074400, A196020, A236106, A237048, A237593, A239050, A239662, A244050, A272027.

T(n,k) = 3*A196020(n,k) = A196020(n,k) + A236106(n,k).

cp's OEIS Frontend

A244370 Total number of toothpicks after n-th stage in the toothpick structure of the symmetric representation of sigma in the four quadrants.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A239052 Sum of divisors of 4*n-2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A239053 Sum of divisors of 4*n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A244971 Number of regions in the symmetric representation of sigma(n) on the four quadrants.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A340426 Triangle read by rows: T(n,k) = A000203(n-k+1)*A002865(k-1), 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A215947 Difference between the sum of the even divisors and the sum of the odd divisors of 2n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A239662 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers A017113 interleaved with k-1 zeros, and the first element of column k is in row k(k+1)/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples