A008590 -id:A008590 - cp's OEIS frontend

A217873 a(n) = 4n(n^2 + 2)/3.

0, 4, 16, 44, 96, 180, 304, 476, 704, 996, 1360, 1804, 2336, 2964, 3696, 4540, 5504, 6596, 7824, 9196, 10720, 12404, 14256, 16284, 18496, 20900, 23504, 26316, 29344, 32596, 36080, 39804, 43776, 48004, 52496, 57260, 62304, 67636, 73264, 79196, 85440, 92004

Offset: 0

M. F. Hasler, Oct 13 2012

Occurs as 4th column in the table A142978 of figurate numbers for n-dimensional cross polytope.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A006527, A008412, A008590, A022144, A082138, A112087, A142978, A174794, A292022.

[4*n*(n^2+2)/3: n in [0..45]]; // Vincenzo Librandi, Nov 08 2012

Table[4n(n^2 + 2)/3, {n, 0, 39}] (* Alonso del Arte, Oct 22 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,4,16,44},50] (* Harvey P. Dale, Mar 16 2015 *)

makelist(4*n*(n^2+2)/3, n, 0, 41); /* Martin Ettl, Oct 15 2012 */

PARI
```
a(n)=(n^2+2)*n/3*4
```

a(n) = 4*A006527(n).
From Peter Luschny, Oct 14 2012: (Start)
a(n) = A008412(n)/2.
a(n) = A174794(n+1) - 1.
First differences are in A112087.
Second differences are in A008590 and A022144.
Binomial transformation of (a(n), n > 0) is A082138.  (End)
G.f.: 4*x*(1 + x^2)/(x - 1)^4. - R. J. Mathar, Oct 15 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), a(0)=0, a(1)=4, a(2)=16, a(3)=44. - Harvey P. Dale, Mar 16 2015
From Elmo R. Oliveira, Aug 09 2025: (Start)
E.g.f.: 4*exp(x)*x*(3 + 3*x + x^2)/3.
a(n) = A292022(n)/3. (End)

A253513 The characteristic function of the multiples of eight.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Mikael Aaltonen, Jan 03 2015

Period 8: repeat [1, 0, 0, 0, 0, 0, 0, 0].

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,1).

Cf. A008590 (multiples of 8), A010877, A014025, A168181, A244413.

Table[Boole[IntegerQ[n/8]], {n, 0, 127}] (* Michael De Vlieger, Jan 03 2015, after Alonso del Arte at A121262 *)
LinearRecurrence[{0,0,0,0,0,0,0,1},{1,0,0,0,0,0,0,0},120] (* or *) PadRight[ {},120,{1,0,0,0,0,0,0,0}] (* Harvey P. Dale, Apr 07 2017 *)
Array[ Boole[ Mod[#, 8] == 0] &, 105, 0] (* Robert G. Wilson v, Oct 08 2017 *)

A253513(n) = !(n%8); \\ Antti Karttunen, Oct 08 2017

def A253513(n): return int(not(n&7)) # Chai Wah Wu, Jul 09 2022

a(n) = floor(n/8) - floor((n-1)/8).
a(n) = sin((sin(Pi*(n+1)/2)^2)*Pi*(n+2)/4)/2 + (sin(Pi*(n+1)/2)^2)/4 + sin(Pi*(n+1)/2)/4.
a(n) = abs(A014025(n)).
From Alois P. Heinz, Jan 03 2015: (Start)
a(n) = 1 - A168181(n).
G.f.: 1/(1-x^8). (End)

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).

A017066 a(n) = (8*n)^2.

Original entry on oeis.org

0, 64, 256, 576, 1024, 1600, 2304, 3136, 4096, 5184, 6400, 7744, 9216, 10816, 12544, 14400, 16384, 18496, 20736, 23104, 25600, 28224, 30976, 33856, 36864, 40000, 43264, 46656, 50176, 53824, 57600, 61504, 65536, 69696, 73984, 78400, 82944, 87616, 92416, 97344, 102400

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008590, A016802, A152691, A244082.

[(8*n)^2: n in [0..35]]; // Vincenzo Librandi, Jul 10 2011

LinearRecurrence[{3, -3, 1}, {0, 64, 256}, 50] (* Vincenzo Librandi, Feb 10 2012 *)

a(n)=(8*n)^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: -64*x*(1+x)/(x-1)^3. - R. J. Mathar, Jul 14 2016
a(n) = A000290(8*n) = A008590(n)^2 = A000290(A008590(n)).
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/384.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/768.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/8)/(Pi/8).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/8)/(Pi/8) = 4*sqrt(2-sqrt(2))/Pi. (End)
From Elmo R. Oliveira, Dec 06 2024: (Start)
E.g.f.: 64*exp(x)*x*(1 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = n*A152691(n) = 2*A244082(n) = A016802(2*n). (End)

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 *)

A326296 Triangle of numbers T(n,k) = 2floor(k/2)(n-k) + ceiling((k-1)^2/2), 1<=k<=n.

Original entry on oeis.org

0, 0, 1, 0, 3, 2, 0, 5, 4, 5, 0, 7, 6, 9, 8, 0, 9, 8, 13, 12, 13, 0, 11, 10, 17, 16, 19, 18, 0, 13, 12, 21, 20, 25, 24, 25, 0, 15, 14, 25, 24, 31, 30, 33, 32, 0, 17, 16, 29, 28, 37, 36, 41, 40, 41, 0, 19, 18, 33, 32, 43, 42, 49, 48, 51, 50, 0, 21, 20, 37, 36, 49, 48, 57, 56, 61, 60, 61

Offset: 1

M. Ryan Julian Jr., Sep 10 2019

T(n,k) gives the maximum number of inversions in a permutation on n symbols containing a single k-cycle and (n-k) other fixed points.
T(n,n) = A000982(n).
T(n,n-1) = A097063(n).

			Triangle begins:
0;
0, 1;
0, 3, 2;
0, 5, 4, 5;
0, 7, 6, 9, 8;
0, 9, 8, 13, 12, 13;
0, 11, 10, 17, 16, 19, 18;
0, 13, 12, 21, 20, 25, 24, 25;
0, 15, 14, 25, 24, 31, 30, 33, 32;
0, 17, 16, 29, 28, 37, 36, 41, 40, 41;
0, 19, 18, 33, 32, 43, 42, 49, 48, 51, 50;
0, 21, 20, 37, 36, 49, 48, 57, 56, 61, 60, 61;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Diagonals give A000982, A097063, A326657, A326658.
Columns give A000004, A005408, A005843, A016813, A008586, A016921, A008588, A017077, A008590.
Row sums give A000330.

T(n,k) = {2*floor(k/2)*(n-k) + ceil((k-1)^2/2)} \\ Andrew Howroyd, Sep 10 2019

T(n,k) = 2*floor(k/2)*(n-k) + ceiling((k-1)^2/2).

T(n,k) = 2*floor(k/2)*(n-k) + binomial(k,2) - ceiling(k/2) + 1.

A365886 Numbers k whose least prime divisor is smaller than its exponent in the prime factorization of k.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 81, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 243, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 405, 408, 416

Offset: 1

Amiram Eldar, Sep 22 2023

First differs from A185359 at n = 22.
Numbers k such that A020639(k) < A051904(k).
The asymptotic density of terms with least prime factor prime(n) (within all the positive integers) is d(n) = (1/prime(n)^(prime(n)+1)) * Product_{k=1..(n-1)} (1-1/prime(k)). For example, for n = 1, 2, 3, 4 and 5, d(n) = 1/8, 1/162, 1/46875, 4/86472015 and 8/109844993185235.
The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.13119421909731920416... .

			8 = 2^3 is a term since its least prime factor, 2, is smaller than its exponent, 3.

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

Cf. A020639, A051904.
Subsequences: A008590 \ {0}, A365887, A365888.
Subsequence of A185359.

q[n_] := Less @@ FactorInteger[n][[1]]; Select[Range[2, 420], q]

is(n) = {my(f = factor(n)); n > 1 && f[1, 1] < f[1, 2];}

A017067 a(n) = (8*n)^3.

Original entry on oeis.org

0, 512, 4096, 13824, 32768, 64000, 110592, 175616, 262144, 373248, 512000, 681472, 884736, 1124864, 1404928, 1728000, 2097152, 2515456, 2985984, 3511808, 4096000, 4741632, 5451776, 6229504, 7077888

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A008590.

[(8*n)^3: n in [0..35]]; // Vincenzo Librandi, Jul 10 2011

G.f.: 512*x*(1 + 4*x + x^2)/(-1+x)^4. - R. J. Mathar, Jun 24 2009

a(n) = A008590(n)^3.

A092100 Smallest number of 1's in binary representations of primes between 2^n and 2^(n+1) is 4.

Original entry on oeis.org

25, 32, 40, 43, 48, 56, 58, 64, 96, 104, 112, 120, 128, 134, 140, 145, 152, 160, 176, 185, 192, 208, 212, 224, 235, 240, 244, 248, 252, 256, 264, 272, 280, 286, 288, 292, 302, 304, 308, 320, 326, 332, 348, 356, 360, 384, 392, 394, 400

Offset: 1

Robert G. Wilson v, Feb 19 2004

nonn

Where 4 appears in A091935.
This sequence differs from multiples of 8 (A008590) very little but significantly; even fewer are odd.
Essentially the same as A081504. - R. J. Mathar, Sep 08 2008

Cf. A091935, A091936.

Compute the second line of the Mathematica code for A091936, then Do[ If[ Count[ IntegerDigits[ f[n], 2], 1] == 4, Print[n]], {n, 1, 400}] (* Robert G. Wilson v, Feb 19 2004 *)

A144449 a(n) = 4(4 + 9n^2 + 15*n).

Original entry on oeis.org

16, 112, 280, 520, 832, 1216, 1672, 2200, 2800, 3472, 4216, 5032, 5920, 6880, 7912, 9016, 10192, 11440, 12760, 14152, 15616, 17152, 18760, 20440, 22192, 24016, 25912, 27880, 29920, 32032, 34216, 36472, 38800, 41200, 43672, 46216, 48832, 51520, 54280, 57112

Offset: 0

Paul Curtz, Oct 06 2008

A decimation: A061039(6n+5).

a(n) mod 9 = period 3: repeat 7,4,1 = A070403(n+1).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A061039, A070403.

Subsequence of A008590.

[36*n^2 + 60*n + 16: n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

Table[36n^2+60n+16,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{16,112,280},40] (* Harvey P. Dale, Apr 04 2020 *)

a(n)=36*n^2+60*n+16 \\ Charles R Greathouse IV, Jun 17 2017

[(6*n+5)^2 - 9 for n in (0..40)] # G. C. Greubel, Mar 06 2022

a(n) = a(n-1) + 24*(3*n+1) = a(n-1) + 72*n + 24, a(0)=16.
A061039(6n+2) = A061039(6n-4) + 24*(3n+1) = a(6n-4) + 72*n + 24, a(2)=16.
From G. C. Greubel, Mar 06 2022: (Start)
G.f.: 8*(2 + 8*x - x^2)/(1-x)^3.
E.g.f.: 4*(4 + 24*x + 9*x^2)*exp(x). (End)
From Amiram Eldar, Mar 11 2022: (Start)
Sum_{n>=0} 1/a(n) = 1/12.
Sum_{n>=0} (-1)^n/a(n) = Pi/(18*sqrt(3)) + log(2)/18 - 1/12. (End)

Edited by Charles R Greathouse IV, Jul 25 2010

cp's OEIS Frontend

A217873 a(n) = 4*n*(n^2 + 2)/3.

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A253513 The characteristic function of the multiples of eight.

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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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A017066 a(n) = (8*n)^2.

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

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A326296 Triangle of numbers T(n,k) = 2*floor(k/2)*(n-k) + ceiling((k-1)^2/2), 1<=k<=n.

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A365886 Numbers k whose least prime divisor is smaller than its exponent in the prime factorization of k.

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A017067 a(n) = (8*n)^3.

A217873 a(n) = 4n(n^2 + 2)/3.

A326296 Triangle of numbers T(n,k) = 2floor(k/2)(n-k) + ceiling((k-1)^2/2), 1<=k<=n.

A144449 a(n) = 4(4 + 9n^2 + 15*n).