A272211 -id:A272211 - cp's OEIS frontend

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.

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).

A272173 Product of the sum of the divisors of n and the sum of the divisors of n-th prime.

Original entry on oeis.org

3, 12, 24, 56, 72, 168, 144, 300, 312, 540, 384, 1064, 588, 1056, 1152, 1674, 1080, 2418, 1360, 3024, 2368, 2880, 2016, 5400, 3038, 4284, 4160, 6048, 3300, 8208, 4096, 8316, 6624, 7560, 7200, 13832, 6004, 9840, 9408, 15660, 7560, 17472, 8448, 16296, 15444, 14400, 10176, 27776

Offset: 1

Omar E. Pol, Apr 26 2016

nonn

Numbers that occur twice in the sequence include 7560, 816000, 2709504, 31752000. Are there infinitely many? Does any number occur more than twice? - Robert Israel, Sep 12 2018

			For n = 9 the sum of the divisors of 9 is 1 + 3 + 9 = 13, and the 9th prime is 23, and the sum of the divisors of 23 is 1 + 23 = 24, and 13*24 = 312, so a(9) = 312.
On the other hand 9*23 = 207, and the sum of the divisors of 207 is 1 + 3 + 9 + 23 + 69 + 207 = 312, so a(9) = 312.

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

Cf. A000040, A000203, A008864, A033286, A272211.

[SumOfDivisors(n)*SumOfDivisors(NthPrime(n)): n in [1..50]]; // Vincenzo Librandi, Sep 13 2018

f:= n -> numtheory:-sigma(n)*(1+ithprime(n)):
map(f, [$1..100]); # Robert Israel, Sep 12 2018

Table[DivisorSigma[1, n]*DivisorSigma[1, Prime[n]], {n, 1, 50}] (* G. C. Greubel, Apr 27 2016 *)

a(n) = sigma(n)*sigma(prime(n)); \\ Michel Marcus, Apr 27 2016

a(n) = sigma(n)*sigma(prime(n)) = sigma(n)*(1 + prime(n)) = A000203(n)*(1 + A000040(n)) = A000203(n)*A008864(n).
a(n) = sigma(n*prime(n)) = A000203(n*A000040(n)) = A000203(A033286(n)).
a(n) = A000203(n) + A272211(n).

A272400 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th noncomposite number and the sum of the divisors of k, n>=1, k>=1.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 5, 9, 8, 7, 7, 15, 12, 14, 6, 11, 21, 20, 21, 12, 12, 13, 33, 28, 35, 18, 24, 8, 17, 39, 44, 49, 30, 36, 16, 15, 19, 51, 52, 77, 42, 60, 24, 30, 13, 23, 57, 68, 91, 66, 84, 40, 45, 26, 18, 29, 69, 76, 119, 78, 132, 56, 75, 39, 36, 12, 31, 87, 92, 133, 102, 156, 88, 105, 65, 54, 24, 28

Offset: 1

Omar E. Pol, Apr 28 2016

			The corner of the square array begins:
1,   3,   4,   7,   6,  12,   8,  15,  13,  18...
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...
...

Rows 1-3: A000203, A074400, A272027.
Columns 1-2: A008578, A112773.
The diagonal 2, 9, 20... is A272211, the main diagonal of A272214.
Cf. A272173.

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

T(n,k) = A008578(n)*A000203(k), n>=1, k>=1.

T(n,k) = A272214(n-1,k), n>=2.

cp's OEIS Frontend

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.

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A272173 Product of the sum of the divisors of n and the sum of the divisors of n-th prime.

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A272400 Square array read by antidiagonals upwards in which T(n,k) is the product of the n-th noncomposite number and the sum of the divisors of k, n>=1, k>=1.

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