A074400 -id:A074400 - cp's OEIS frontend

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.

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).

A272214 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th prime and the sum of the divisors of k, n >= 1, k >= 1.

Original entry on oeis.org

2, 3, 6, 5, 9, 8, 7, 15, 12, 14, 11, 21, 20, 21, 12, 13, 33, 28, 35, 18, 24, 17, 39, 44, 49, 30, 36, 16, 19, 51, 52, 77, 42, 60, 24, 30, 23, 57, 68, 91, 66, 84, 40, 45, 26, 29, 69, 76, 119, 78, 132, 56, 75, 39, 36, 31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24, 37, 93, 116, 161, 114, 204, 104, 165, 91, 90, 36, 56

Offset: 1

Omar E. Pol, Apr 28 2016

From Omar E. Pol, Dec 21 2021: (Start)
Also triangle read by rows: T(n,j) = A000040(n-j+1)*A000203(j), 1 <= j <= n.
For a visualization of T(n,j) first consider a tower (a polycube) in which the terraces are the symmetric representation of sigma(j), for j = 1 to n, starting from the top, and the heights of the terraces are the first n prime numbers respectively starting from the base. Then T(n,j) can be represented with a set of A237271(j) right prisms of height A000040(n-j+1) since T(n,j) is also the total number of cubes that are exactly below the parts of the symmetric representation of sigma(j) in the tower.
The sum of the n-th row of triangle is A086718(n) equaling the volume of the tower whose largest side of the base is n and its total height is the n-th prime.
The tower is an member of the family of the stepped pyramids described in A245092 and of the towers described in A221529. That is an infinite family of symmetric polycubes whose volumes represent the convolution of A000203 with any other integer sequence. (End)

			The corner of the square array begins:
   2,  6,   8,  14,  12,  24,  16,  30,  26,  36, ...
   3,  9,  12,  21,  18,  36,  24,  45,  39,  54, ...
   5, 15,  20,  35,  30,  60,  40,  75,  65,  90, ...
   7, 21,  28,  49,  42,  84,  56, 105,  91, 126, ...
  11, 33,  44,  77,  66, 132,  88, 165, 143, 198, ...
  13, 39,  52,  91,  78, 156, 104, 195, 169, 234, ...
  17, 51,  68, 119, 102, 204, 136, 255, 221, 306, ...
  19, 57,  76, 133, 114, 228, 152, 285, 247, 342, ...
  23, 69,  92, 161, 138, 276, 184, 345, 299, 414, ...
  29, 87, 116, 203, 174, 348, 232, 435, 377, 522, ...
  ...
From _Omar E. Pol_, Dec 21 2021: (Start)
Written as a triangle the sequence begins:
   2;
   3,  6;
   5,  9,  8;
   7, 15, 12,  14;
  11, 21, 20,  21,  12;
  13, 33, 28,  35,  18,  24;
  17, 39, 44,  49,  30,  36, 16;
  19, 51, 52,  77,  42,  60, 24,  30;
  23, 57, 68,  91,  66,  84, 40,  45, 26;
  29, 69, 76, 119,  78, 132, 56,  75, 39, 36;
  31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24;
...
Row sums give A086718. (End)

Ivan Neretin, Table of n, a(n) for n = 1..8128

Rows 1-4 of the square array: A074400, A272027, A274535, A319527.
Columns 1-5 of the square array: A000040, A001748, A001749, A138636, A272470.
Main diagonal of the square array gives A272211.
Cf. A086718 (antidiagonal sums of the square array, row sums of the triangle).
Cf. A000203, A221529, A237270, A237271, A237593, A245092, A272173, A272400, A274824.

Table[Prime[#] DivisorSigma[1, k] &@(n - k + 1), {n, 12}, {k, n}] // Flatten (* Michael De Vlieger, Apr 28 2016 *)

T(n,k) = prime(n)*sigma(k) = A000040(n)*A000203(k), n >= 1, k >= 1.

T(n,k) = A272400(n+1,k).

A294015 Sum of the even divisors of 2n, minus the (n-1)st odd number.

Original entry on oeis.org

1, 3, 3, 7, 3, 13, 3, 15, 9, 17, 3, 33, 3, 21, 19, 31, 3, 43, 3, 45, 23, 29, 3, 73, 13, 33, 27, 57, 3, 85, 3, 63, 31, 41, 27, 111, 3, 45, 35, 101, 3, 109, 3, 81, 67, 53, 3, 153, 17, 87, 43, 93, 3, 133, 35, 129, 47, 65, 3, 217, 3, 69, 83, 127, 39, 157, 3, 117, 55, 149, 3, 247, 3, 81, 99, 129, 39, 181, 3, 213, 81, 89, 3, 281

Offset: 1

Omar E. Pol, Oct 28 2017

nonn

a(n) = 3 if and only if n is prime.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Partial sums give A294016.
Cf. A000203, A013661, A005408, A074400, A235796, A294017.
Cf. A001065, A091818.

a[n_] := 2*(DivisorSigma[1, n] - n) + 1; Array[a, 100] (* Amiram Eldar, Mar 30 2024 *)

a(n) = 2*sigma(n) - 2*n + 1; \\ Michel Marcus, Oct 29 2017

a(n) = A074400(n) - A005408(n-1) = 2*A000203(n) - 2*n + 1 = A000203(n) - A235796(n).
Sum_{k=1..n} a(k) = (Pi^2/6 - 1) * n^2 + O(n*log(n)). - Amiram Eldar, Mar 30 2024
a(n) = 2*A001065(n) + 1 = A091818(n) + 1. - Omar E. Pol, Dec 01 2024

A171256 Numbers n such that sigma(n) = 10*phi(n) (where sigma=A000203, phi=A000010).

Original entry on oeis.org

168, 270, 570, 2376, 2436, 5016, 6426, 7110, 13566, 15834, 34452, 58520, 62568, 72732, 75210, 113832, 126882, 168756, 169218, 191862, 199368, 223938, 240312, 280488, 308568, 321468, 420888, 449442, 472758, 661848, 673608, 776736, 848540, 854496, 907236

Offset: 1

M. F. Hasler, Mar 19 2010

nonn

If n is in this sequence, then for any prime p not dividing n, sigma(np) - 10*phi(np) = 2*sigma(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Jud McCranie, terms 1..1000 from Donovan Johnson)
Kevin A. Broughan and Daniel Delbourgo, On the Ratio of the Sum of Divisors and Euler’s Totient Function I, Journal of Integer Sequences, Vol. 16 (2013), Article 13.8.8.
Kevin A. Broughan and Qizhi Zhou, On the Ratio of the Sum of Divisors and Euler's Totient Function II, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.2.

Cf. A062699, A068391, A074400, A068390, A136547, A104900, A136540, A104901, A163667, A171257, A104902, A171258, A171259, A171260, A104903.

Select[Range[10^6], DivisorSigma[1, #] == 10 * EulerPhi[#] &] (* Amiram Eldar, Dec 04 2019 *)

for(k=1,10^6, sigma(k) - 10*eulerphi(k) || print1(k", "));

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).

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).

A091818 Sum of the even proper divisors of 2n. Sum of the even divisors of 2n that are less than 2n.

Original entry on oeis.org

0, 2, 2, 6, 2, 12, 2, 14, 8, 16, 2, 32, 2, 20, 18, 30, 2, 42, 2, 44, 22, 28, 2, 72, 12, 32, 26, 56, 2, 84, 2, 62, 30, 40, 26, 110, 2, 44, 34, 100, 2, 108, 2, 80, 66, 52, 2, 152, 16, 86, 42, 92, 2, 132, 34, 128, 46, 64, 2, 216, 2, 68, 82, 126, 38, 156, 2, 116

Offset: 1

Mohammad K. Azarian, Mar 07 2004

			The sum of the even divisors of 18 that are less than 18 is 8 = 2+6.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A001065, A032741, A091570, A074400, A173455.

Table[Total[Select[Most[Divisors[2 n]],EvenQ]],{n,70}] (* Harvey P. Dale, Apr 28 2023 *)

a(n) = sumdiv(2*n, d, !(d%2) * d * (d<2*n)); \\ Michel Marcus, Jan 14 2014

from sympy import divisors
def a(n): return sum(d for d in divisors(2*n) if d%2==0) - 2*n
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Oct 30 2017

a(n) = A074400(2n) - 2n. - Michel Marcus, Jan 14 2014
a(n) = Sum_{d|2n, d<2n, d even} d. - Wesley Ivan Hurt, Mar 02 2022
a(n) = 2 * A001065(n). - Alois P. Heinz, Mar 02 2022

More terms from Michel Marcus, Jan 14 2014

A100892 a(n) = (2n-1) XOR (2n+1), bitwise.

Original entry on oeis.org

2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 62, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 126, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 62, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 254, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2, 6, 2, 62, 2, 6, 2, 14, 2, 6, 2, 30, 2, 6, 2, 14, 2

Offset: 1

Reinhard Zumkeller, Jan 10 2005

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006519, A099820, A007088.

Cf. A005408.

a100892 n = (2 * n - 1) `xor` (2 * n + 1)
a100892_list = zipWith xor (tail a005408_list) a005408_list
-- Reinhard Zumkeller, Sep 03 2013

a[n_]:=BitXor[2*n-1,2*n+1]; a/@Range[100] (* Ivan N. Ianakiev, Jul 04 2019 *)

a(n)=4*2^valuation(n,2)-2; \\ Ralf Stephan, Aug 21 2013

def A100892(n): return ((~n& n-1)<<2)+2 # Chai Wah Wu, Jul 07 2022

a(n) = 2 * ((n-1) XOR n) = 2*A038712(n).
a(n) = 4*2^A007814(n) - 2.
Recurrence: a(2n) = 2a(n) + 2, a(2n+1) = 2. - Ralf Stephan, Aug 21 2013
a(n) = A088837(n) - 1. - Filip Zaludek, Dec 10 2016
a(n) = A074400(n)/A000593(n) = 2*A000203(n)/A000593(n). - Ivan N. Ianakiev, Jul 04 2019

A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Sep 22 2018

			The corner of the square array begins:
         A000203 A074400 A272027 A239050 A274535 A274536 A319527 A319528
A000027:       1,      2,      3,      4,      5,      6,      7,      8, ...
A008585:       3,      6,      9,     12,     15,     18,     21,     24, ...
A008586:       4,      8,     12,     16,     20,     24,     28,     32, ...
A008589:       7,     14,     21,     28,     35,     42,     49,     56, ...
A008588:       6,     12,     18,     24,     30,     36,     42,     48, ...
A008594:      12,     24,     36,     48,     60,     72,     84,     96, ...
A008590:       8,     16,     24,     32,     40,     48,     56,     64, ...
A008597:      15,     30,     45,     60,     75,     90,    105,    120, ...
A008595:      13,     26,     39,     52,     65,     78,     91,    104, ...
A008600:      18,     36,     54,     72,     90,    108,    126,    144, ...
...

Index entries for sequences related to sigma(n)

Another version of A274824.
Antidiagonal sums give A175254.
Main diagonal gives A064987.
Row n lists the multiples of A000203(n).
Row 1 is A000027.
Initial zeros should be omitted in the following sequences related to the rows of the array:
Row 2-5: A008585, A008586, A008589, A008588.
Rows 6 and 11: A008594.
Rows 7-9: A008590, A008597, A008595.
Rows 10 and 17: A008600.
Rows 12-13: A135628, A008596.
Rows 14, 15 and 23: A008606.
Rows 16 and 25: A135631.
(Note that in the OEIS there are many other sequences that are also rows of this square array.)
Cf. A000203, A237593, A319526.

T:=Flat(List([1..12],n->List([1..n],k->k*Sigma(n-k+1))));; Print(T); # Muniru A Asiru, Jan 01 2019

with(numtheory): T:=(n,k)->k*sigma(n-k+1): seq(seq(T(n,k),k=1..n),n=1..12); # Muniru A Asiru, Jan 01 2019

Table[k DivisorSigma[1, #] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

A340583 Triangle read by rows: T(n,k) = A002865(n-k)*A000203(k), 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 15 2021

T(n,k) is the total number of cubic cells added at n-th stage to the right prisms whose bases are the parts of the symmetric representation of sigma(k) in the polycube described in A221529.

Partial sums of column k gives the column k of A221529.

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

Row sums give A138879.
Column 1 gives A002865.
Diagonals 1, 3 and 4 give A000203.
Diagonal 2 gives A000004.
Diagonals 5 and 6 give A074400.
Diagonals 7 and 8 give A239050.
Diagonal 9 gives A319527.
Diagonal 10 gives A319528.
Cf. A221529 (partial column sums).
Cf. A340426 (mirror).
Cf. A135010, A221530, A245095, A245099, A339278.

A340583[n_, k_] := (PartitionsP[n - k] - PartitionsP[(n - k) - 1])*
   DivisorSigma[1, k];
Table[A340583[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Robert P. P. McKone, Jan 25 2021 *)

cp's OEIS Frontend

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