A272027 -id:A272027 - 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).

A077087 Numbers k such that sigma(k+1) = 3 * sigma(k).

Original entry on oeis.org

1, 1919, 2759, 11219, 27089, 50039, 58463, 100127, 113831, 115289, 120203, 131879, 148511, 233729, 244319, 308039, 461099, 554063, 596447, 1406303, 1486619, 2285519, 2880989, 5138783, 5369111, 5521619, 5736743, 6621383, 7496279, 7683191, 8571527, 8848619

Offset: 1

Labos Elemer, Oct 31 2002

nonn

			k = 1: sigma(2)/sigma(1) = 3/1 = 3;
k = 9563231: sigma(k+1)/sigma(k) = 31026240/10342080 = 3.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000203, A002961, A067081, A077086, A272027 (3*sigma(n)).

Do[s=Mod[a=DivisorSigma[1, n+1], b=DivisorSigma[1, n]]; If[Equal[s, 0]&&Equal[a/b, 3], Print[n]], {n, 1, 10000000}]
Position[Partition[DivisorSigma[1,Range[9*10^6]],2,1],?(3#[[1]] == #[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Oct 03 2017 *)

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).

A326181 Numbers n for which sigma(sigma(n)) = 3*sigma(n).

Original entry on oeis.org

54, 56, 87, 95, 276, 308, 429, 446, 455, 501, 581, 611, 158928, 194928, 195072, 199950, 226352, 234608, 236432, 248325, 255678, 263504, 266192, 273050, 275415, 304575, 336903, 341162, 353675, 366575, 369425, 369843, 380463, 386313, 389463, 406565, 411725, 415925, 422303, 447587, 468743, 497333, 500993, 511829, 515267, 519557, 519677

Offset: 1

Antti Karttunen, Jun 16 2019

nonn

Any odd perfect numbers must occur in this sequence, as such numbers must be in the intersection of A000396 and A326051, that is, satisfy both sigma(n) = 2n and sigma(2n) = 6n = 3*2n, thus in combination they must satisfy sigma(sigma(n)) = 3*sigma(n). Note that odd perfect numbers should occur also in A019283.

If, as conjectured, A005820 has 6 terms, then this sequence is finite and has 756 terms. - Giovanni Resta, Jun 17 2019

Giovanni Resta, Table of n, a(n) for n = 1..756 (first 169 terms from Antti Karttunen)
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A000396, A005820, A019283, A051027, A087943, A272027 (3*sigma(n)), A326051.

Subsequence of A066961.

isA326181(n) = { my(s=sigma(n)); (sigma(s)==3*s); };

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 *)

A175875 Numbers k such that sigma(k+3) = 3*sigma(k).

Original entry on oeis.org

3, 30237, 292317, 3116957, 4044037, 6902277, 73442597, 115767957, 137410557, 416776357, 526908197, 541579497, 695350653, 758403357, 1105731897, 1178082357, 1233277917, 1309742557, 1866261597, 1998267037, 2161411557, 2563416237, 2750761437, 2873771997, 2892203997, 3331848517, 3621735037, 3758847117

Offset: 1

Zak Seidov, Oct 06 2010

nonn

a(1) = A175874(3).

Giovanni Resta, Table of n, a(n) for n = 1..180 (terms < 10^12)

Cf. A000203, A002961, A175874, A272027 (3*sigma(n)).

[n: n in [1..10^8] | SumOfDivisors(n+3) eq 3*SumOfDivisors(n)]; // Vincenzo Librandi, Jul 28 2017

isok(n) = sigma(n+3) == 3*sigma(n); \\ Michel Marcus, Oct 19 2013

a(8)-a(21) from Donovan Johnson, Oct 11 2010
a(22)-a(24) from Donovan Johnson, Sep 10 2012
a(25) from Zak Seidov, Jul 07 2013
a(26)-a(28) from Chai Wah Wu, Jul 27 2017

A217791 Numbers k such that sigma(k) = 3*sigma(k+1).

Original entry on oeis.org

180, 12000, 30996, 47940, 66780, 102816, 128040, 234300, 494088, 712272, 1133088, 1408212, 1623072, 1692768, 1896336, 1925196, 2024760, 2388720, 2529090, 2836008, 3423120, 3724320, 3822360, 4628760, 4750920, 7219608, 7359912, 7603488, 7749060

Offset: 1

Paolo P. Lava, Mar 25 2013

nonn

			47940 is in the sequence because sigma(47940)=145152, sigma(47941)=48384, and 145152=3*48384.
7749060 is in the sequence because sigma(7749060)=24192000, sigma(7749061)=8064000, and 24192000=3*8064000.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000203, A002961, A058073, A067081, A077087, A163193, A272027 (3*sigma(n)).

[n: n in [1..10^7] | SumOfDivisors(n) eq 3*SumOfDivisors(n+1)]; // Bruno Berselli, Mar 25 2013

A217791:=proc(q) local n;
for n from 1 to q do if sigma(n)=3*sigma(n+1) then print(n); fi; od; end:
A217791(10^10);

Position[Partition[DivisorSigma[1,Range[78*10^5]],2,1],?(#[[1]] == 3#[[2]]&), {1},Heads->False]//Flatten (* _Harvey P. Dale, Oct 17 2016 *)

More terms from Bruno Berselli, Mar 25 2013

A245774 Numbers k that divide 3*sigma(k).

Original entry on oeis.org

1, 3, 6, 12, 28, 84, 120, 234, 270, 496, 672, 1080, 1488, 1638, 6048, 6552, 8128, 24384, 30240, 32760, 35640, 199584, 435708, 523776, 2142720, 2178540, 4713984, 12999168, 18506880, 23569920, 33550336, 36197280, 45532800

Offset: 1

Jaroslav Krizek, Aug 26 2014

nonn

Numbers k that divide 3*A000203(k).
Supersequence of A007691 and A245775.
Union of A007691 and 3*A227303. - Robert Israel, Aug 26 2014

			Number 12 is in the sequence because 12 divides 3*sigma(12) = 3*28.

Cf. A000203 (sum of divisors), A007691 (multiply-perfect numbers).
Cf. A227303 (n divides sigma(3n)), A245775 (denominator(sigma(n)/n) = 3).
Cf. A272027 (3*sigma(n)).

[n: n in [1..3000000] | Denominator(3*(SumOfDivisors(n))/n) eq 1]

select(n -> 3*numtheory:-sigma(n) mod n = 0, [$1..10^6]); # Robert Israel, Aug 26 2014

a245774[n_Integer] := Select[Range[n], Divisible[3*DivisorSigma[1, #], #] == True &]; a245774[10^7] (* Michael De Vlieger, Aug 27 2014 *)

for(n=1,10^9,if((3*sigma(n))%n==0,print1(n,", "))) \\ Derek Orr, Aug 26 2014

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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A077087 Numbers k such that sigma(k+1) = 3 * sigma(k).

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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.

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

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A326181 Numbers n for which sigma(sigma(n)) = 3*sigma(n).

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PARI

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

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GAP

Maple

Mathematica

A175875 Numbers k such that sigma(k+3) = 3*sigma(k).

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Magma

PARI

Extensions

A217791 Numbers k such that sigma(k) = 3*sigma(k+1).

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Magma

Maple

Mathematica