A017113 -id:A017113 - cp's OEIS frontend

A329066 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( (Sum_{j=0..n} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n} y^(2j+1)+1/y^(2j+1)) - (Sum_{j=0..n-1} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n-1} y^(2j+1)+1/y^(2j+1)) )^(2*k).

1, 4, 1, 36, 12, 1, 400, 588, 20, 1, 4900, 49440, 2100, 28, 1, 63504, 5187980, 423440, 4956, 36, 1, 853776, 597027312, 117234740, 1751680, 9540, 44, 1, 11778624, 71962945824, 36938855520, 907687900, 5101200, 16236, 52, 1

Offset: 0

Seiichi Manyama, Nov 03 2019

T(n,k) is the number of (2*k)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 2*n+1).

T(n,k) is the constant term in the expansion of (Sum_{j=0..2*n+1} (x^j + 1/x^j)*(y^(2*n+1-j) + 1/y^(2*n+1-j)) - x^(2*n+1) - 1/x^(2*n+1) - y^(2*n+1) - 1/y^(2*n+1))^(2*k).

			Square array begins:
   1,  4,   36,     400,       4900, ...
   1, 12,  588,   49440,    5187980, ...
   1, 20, 2100,  423440,  117234740, ...
   1, 28, 4956, 1751680,  907687900, ...
   1, 36, 9540, 5101200, 4190017860, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened
Wikipedia, Taxicab geometry.

Columns k=0-1 give A000012, A017113.
Rows n=0-2 give A002894, A329024, A329067.
Main diagonal gives A342964.
Cf. A005408, A328718, A329074, A329078.

{T(n, k) = polcoef(polcoef((sum(j=0, 2*n+1, (x^j+1/x^j)*(y^(2*n+1-j)+1/y^(2*n+1-j)))-x^(2*n+1)-1/x^(2*n+1)-y^(2*n+1)-1/y^(2*n+1))^(2*k), 0), 0)}

f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
T(n, k) = sum(j=0, 2*k, (-1)^j*binomial(2*k, j)*polcoef(f(n)^j*f(n-1)^(2*k-j), 0)^2)

See the second code written in PARI.

A047457 Numbers that are congruent to {3, 4} mod 8.

Original entry on oeis.org

3, 4, 11, 12, 19, 20, 27, 28, 35, 36, 43, 44, 51, 52, 59, 60, 67, 68, 75, 76, 83, 84, 91, 92, 99, 100, 107, 108, 115, 116, 123, 124, 131, 132, 139, 140, 147, 148, 155, 156, 163, 164, 171, 172, 179, 180, 187, 188, 195, 196, 203, 204, 211, 212, 219, 220, 227

Offset: 1

N. J. A. Sloane

Union of A017101 and A017113. - Michel Marcus, Feb 25 2014

Numbers whose binary reflected Gray code (A014550) has a single trailing zero. - Amiram Eldar, May 17 2021

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A000120, A014550, A017101, A017113, A047456 (superset), A063787.

A047457:=n->(-5 - 3*(-1)^n + 8*n)/2; seq(A047457(n), n=1..100); # Wesley Ivan Hurt, Mar 04 2014

Table[(-5 - 3*(-1)^n + 8*n)/2, {n, 100}] (* Wesley Ivan Hurt, Mar 04 2014 *)
Flatten[Table[8n + {3, 4}, {n, 0, 29}]] (* Alonso del Arte, Mar 04 2014 *)

a(n) = 8*n - a(n-1) - 9 (with a(1) = 3). - Vincenzo Librandi, Aug 06 2010
G.f.: x*(3+x+4*x^2)/((1-x)^2*(1+x)). - Colin Barker, May 13 2012
a(n) = (-5 - 3*(-1)^n + 8*n)/2. - Colin Barker, May 14 2012
A000120(a(n)-1) = A000120(a(n)+1) = A063787(n). - Ilya Lopatin and Juri-Stepan Gerasimov, Feb 25 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/16 + log(2)/4 - sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 18 2021

A374933 Maximum number of squares covered (i.e., attacked) by 2 independent (i.e., non-attacking) queens on an n X n chessboard.

Original entry on oeis.org

9, 15, 23, 30, 37, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 308, 316, 324, 332, 340, 348, 356, 364, 372, 380, 388, 396, 404, 412, 420

Offset: 3

John King, Jul 24 2024

It is not possible to place two non-attacking queens on a 1 X 1 or 2 X 2 chessboard.

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A017113, A047461 (case for one queen).

a(n) = 8*n - 20 for n >= 8.

G.f.: x^3*(9 - 3*x + 2*x^2 - x^3 + x^6)/(1 - x)^2. - Stefano Spezia, Jul 25 2024

A337940 Triangle read by rows: T(n, k) = T(n+2) - T(n-k), with the triangular numbers T = A000217, for n >= 1, k = 1, 2, ..., n.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 15, 18, 20, 21, 18, 22, 25, 27, 28, 21, 26, 30, 33, 35, 36, 24, 30, 35, 39, 42, 44, 45, 27, 34, 40, 45, 49, 52, 54, 55, 30, 38, 45, 51, 56, 60, 63, 65, 66, 33, 42, 50, 57, 63, 68, 72, 75, 77, 78, 36, 46, 55, 63, 70, 76, 81, 85, 88, 90, 91

Offset: 1

Wolfdieter Lang, Nov 23 2020

This number triangle results from the array A(n, m) = T(n+m+1) - T(n-1), with T = A000217, for n, m >= 1. For this array see the example by Bob Selcoe, in A111774 (but with rows continued). The present triangle is obtained by reading the array by upwards antidiagonals: T(n, k) = A(n+1-k, k). See also the Jul 09 2019 comment by Ralf Steiner with the formula c_k(n) (rows k >= 1, columns n >= 3), rewritten for A(n, m) = (m+2)*(2*n+m+1)/2, leading to T(n, k) = (k+2)*(2*n-k+3)/2.
Therefore this triangle is related to the problem of giving the numbers which are sums of at least three consecutive positive integers given as sequence A111774. It allows us to find the multiplicities for the numbers of A111774. They are given in A338428(n).
To obtain the multiplicity for number N (>= 6) from A111774 one has to consider only the triangle rows n = 1, 2, ..., floor((N-3)/3).
The row reversed triangle, considered by Bob Selcoe in A111774, is T(n, n-k+1) = T(n+2) - T(k-1), for n >= 1, and k=1, 2, ..., n.
This triangle contains no odd prime numbers and no exact powers 2^m, for m >= 0. This can be seen by considering the diagonal sequences D(d, k), for d >= 1, k >= 1 or the row sequences of the array A(n, m), for n >= 1 and m >= 1. The result is A(r+1, s-2) = s*(s + 2*r + 1)/2, for r >= 0 and s >= 3 (from the g.f. of the diagonals of T given below). This is also given in the Jul 09 2019 comment by Ralf Steiner in A111774. Therefore A(r+1, s-2) is a product of two numbers >= 2, hence not a prime. And in both cases (i) s/2 integer or (ii) (s + 2*r + 1)/2 integer not both numbers can be powers of 2 by simple parity arguments.
The previous comment means that each T(n, k) has at least one odd prime as a proper divisor.
A number N appears in this triangle, or in A111774, if and only if floor(N/2) - delta(N) >= 1, where delta(N) = A055034(N). For the sequence b(n) := floor(n/2) - delta(n), for n >= 2, see A219839(n), b(1) = -1. See a W. Lang comment in A111774 for the proof.

			The triangle T(n, k) begins:
n \ k  1  2  3  4  5   6   7   8   9  10  11  12  13  14  15 ...
1:     6
2:     9 10
3:    12 14 15
4:    15 18 20 21
5:    18 22 25 27 28
6:    21 26 30 33 35  36
7:    24 30 35 39 42  44  45
8:    27 34 40 45 49  52  54  55
9:    30 38 45 51 56  60  63  65  66
10:   33 42 50 57 63  68  72  75  77  78
11:   36 46 55 63 70  76  81  85  88  90  91
12:   39 50 60 69 77  84  90  95  99 102 104 105
13:   42 54 65 75 84  92  99 105 110 114 117 119 120
14:   45 58 70 81 91 100 108 115 121 126 130 133 135 136
15:   48 62 75 87 98 108 117 125 132 138 143 147 150 152 153
...
N = 15 appears precisely twice from the sums 4+5+6 = A(4, 1) = T(4, 1), and (1+2+3)+4+5 = A(1, 3) = T(3, 3), i.e., with a sum of 3 and 5 consecutive positive integers.
N = 42 appears three times from the sums 13+14+15 = A(13, 1) = T(13, 1), 9+10+11 +12 = A(9, 2) = T(10, 2), 3+4+5+6+7+8+9 = A(3, 5) = T(7, 5); i.e., 42 can be written as a sum of 3, 4 and 7 consecutive positive integers.

Cf. A055034, A111774, A338428 (multiplicities), A219839.
For columns k = 1, 2, ..., 10 see A008585, A016825, A008587, A016945, A008589, A017113, A008591, A017329, A008593, A017593.
For diagonals d = 1, 2, ..., 10 see A000217, A000096, A055998, A055999, A056000, A056115, A056119 , A056121, A056126, A051942.

Flatten[Table[((n+2)*(n+3)-(n-k)*(n-k+1))/2,{n,11},{k,n}]] (* Stefano Spezia, Nov 24 2020 *)

T(n, k) = ((n+2)*(n+3) - (n-k)*(n-k+1))/2, for n >= 1 and k = 1, 2, ..., n (see the name).
T(n, k) = (k+2)*(2*n-k+3)/2 (factorized).
G.f. columns k = 2*j+1, for j >= 0: Go(j, x) = x^(2*j+1)*(2*j+3)*(j+2 - (j+1)*x)/(1-x)^2,
G.f. columns k = 2*j, for j >= 1: Ge(j, x) = x^(2*j)*(j+1)*(2*j+3 - (2*j+1)*x)/(1-x)^2.
G.f. row polynomials: G(z,x) = z*x*(1 + z*x)^3*{3*(2-z) - (8-3*z)*(z*x) + (3-z)*(z*x)^2}/((1 - z)^2*(1 - (z*x)^2)^3).
G.f. diagonals d >= 1: GD(d, x) = ((d+1)*3 - (5*d+3)*x + (2*d+1)*x^2)/(1-x)^3.
G.f. of GD(d, x): GGD(z,x) = (6-8*x+3*x^2 - (3-3*x+x^2)*z)/((1-x)^3*(1-z)^2).

A365883 Numbers k whose least prime divisor is equal to its exponent in the prime factorization of k.

Original entry on oeis.org

4, 12, 20, 27, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 135, 140, 148, 156, 164, 172, 180, 188, 189, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 297, 300, 308, 316, 324, 332, 340, 348, 351, 356, 364, 372, 380, 388, 396

Offset: 1

Amiram Eldar, Sep 22 2023

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/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, 1/81, 4/46875, 8/28824005 and 16/21968998637047.
The asymptotic density of this sequence is Sum_{n>=1} d(n) = 0.13743128989284883653... .

			4 = 2^2 is a term since its least prime factor, 2, is equal to its exponent.

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

Cf. A020639, A051904.
Subsequence of A100717 and A365889.
Subsequences: A017113, A365884, A365885.

filter:= proc(n) local F;
   F:= sort(ifactors(n)[2],(s,t) -> s[1]Robert Israel, Sep 22 2023

q[n_] := Equal @@ FactorInteger[n][[1]]; Select[Range[2, 400], q]

is(n) = n > 1 && #Set(factor(n)[1,]) == 1;

A374934 Maximum number of squares covered (i.e., attacked) by 4 independent (i.e., nonattacking) queens on an n X n chessboard.

Original entry on oeis.org

16, 25, 36, 49, 62, 76, 92, 104, 120, 136, 152, 168, 184, 200, 216

Offset: 4

John King, Aug 08 2024

			4 X 4:
  x Q x x
  x x x Q
  Q x x x
  x x Q x
5 X 5 there are several arrangements:
  x Q x x x
  x x x x x
  x x x x Q
  Q x x x x
  x x x Q x
6 X 6 and 7 X 7 (add a row and column) pattern as 4 queens knight-1,3 and 1,4 separation (not symmetric):
  . . . . . . .
  x x x x Q x .
  Q x x x x x .
  x x x x x x .
  x x x x x Q .
  x Q x x x x .
8 X 8: queens all knight-1,4 apart;
8 X 8 has 2 o/s;
9 X 9 has 5 o/s;
10 X 10 has 8 o/s;
  o x x x x x x x x o
  x o x x x x x x o x
  x x x Q x x x x x x
  x x x x x x x Q x x
  x x x x x x x x x x
  x x x x x x x x x x
  x x Q x x x x x x x
  x x x x x x Q x x x
  x o x x x x x x o x
  o x x x x x x x x o
beyond 10 X 10, the 4 queens separated as 1,2 knights begins to be the best layout; at 15 X 15, the pattern is clear.
  o x x o o x x x x o o x x o x
  x o x x o x x x x o x x o x x
  x x o x x x x x x x x o x x o
  o x x o x x x x x x o x x o o
  o o x x o x x x x o x x o o o
  x x x x x x Q x x x x x x x x
  x x x x x x x x Q x x x x x x
  x x x x x Q x x x x x x x x x
  x x x x x x x Q x x x x x x x
  o o x x o x x x x o x x o o o
  o x x o x x x x x x o x x o o
  x x o x x x x x x x x o x x o
  x o x x o x x x x o x x o x x
  o x x o o x x x x o o x x o x
  x x o o o x x x x o o o x x o

Column 4 of A376732.

Cf. A017113, A047461, A374933, A375116, A374935, A374936, A374937, A374938.

a(18) added using data from Mia Muessig by Andrew Howroyd, Oct 05 2024

A001533 a(n) = (8n+1)(8*n+7).

Original entry on oeis.org

7, 135, 391, 775, 1287, 1927, 2695, 3591, 4615, 5767, 7047, 8455, 9991, 11655, 13447, 15367, 17415, 19591, 21895, 24327, 26887, 29575, 32391, 35335, 38407, 41607, 44935, 48391, 51975, 55687, 59527, 63495, 67591, 71815, 76167, 80647, 85255, 89991, 94855, 99847, 104967

Offset: 0

N. J. A. Sloane

From Klaus Purath, Aug 18 2022: (Start)
This is A028560(8*n+1), and thus a(n) + 9 is a square. (See formulas)
7 is the only prime number of this sequence in which all odd prime factors occur.
Each prime factor p appears exactly twice in any interval of p consecutive terms. If a(m) and a(n) are within such an interval containing p, then m + n == -1 (mod p). (End)

T. D. Noe, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001539, A004771, A017077, A017113, A028560, A250129.

a[n_] := (8 n + 1)*(8 n + 7); Array[a, 40, 0] (* Amiram Eldar, Sep 08 2022 *)
LinearRecurrence[{3,-3,1},{7,135,391},40] (* Harvey P. Dale, Jan 07 2023 *)

a(n)=(8*n+1)*(8*n+7) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 4*A001539(n) - 5.
a(n) = 128*n + a(n-1) with a(0)=7. - Vincenzo Librandi, Nov 12 2010
Sum_{n>=0} 1/a(n) = (Psi(7/8)-Psi(1/8))/48 = 0.1580099..., see A250129. - R. J. Mathar, May 30 2022 [ = (sqrt(2)+1)*Pi/48. - Amiram Eldar, Sep 08 2022]
From Klaus Purath, Aug 18 2022: (Start)
a(n) = A028560(8*n+1).
a(n) + 9 = ((a(n+1) - a(n-1))/32)^2 = A017113(n)^2.
a(2*n) = (a(n+1) - a(n-1))*n + 7. (End)
From Amiram Eldar, Feb 19 2023: (Start)
a(n) = A017077(n)*A004771(n).
Sum_{n>=0} (-1)^n/a(n) = (cos(Pi/8) * log(cot(Pi/16)) + sin(Pi/8) * log(cot(3*Pi/16)))/12.
Product_{n>=0} (1 - 1/a(n)) = cosec(Pi/8)*cos(sqrt(5/2)*Pi/4).
Product_{n>=0} (1 + 1/a(n)) = cosec(Pi/8)*cos(sqrt(2)*Pi/4). (End)
G.f.: -(7+114*x+7*x^2)/(x-1)^3. - R. J. Mathar, Apr 23 2024
From Elmo R. Oliveira, Oct 25 2024: (Start)
E.g.f.: exp(x)*(7 + 64*x*(2 + x)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A294628 a(n) = 8*(sigma(n) - n + (1/2)).

Original entry on oeis.org

4, 12, 12, 28, 12, 52, 12, 60, 36, 68, 12, 132, 12, 84, 76, 124, 12, 172, 12, 180, 92, 116, 12, 292, 52, 132, 108, 228, 12, 340, 12, 252, 124, 164, 108, 444, 12, 180, 140, 404, 12, 436, 12, 324, 268, 212, 12, 612, 68, 348, 172, 372, 12, 532, 140, 516, 188, 260, 12, 868, 12, 276

Offset: 1

Omar E. Pol, Nov 05 2017

nonn

Muniru A Asiru, Table of n, a(n) for n = 1..2000

Partial sums give A294629.

Cf. A001065, A017113, A235796, A237593, A239050, A294015, A294016, A294017, A294630.

List([1..10^5],n->8*(Sigma(n)-n+(1/2))); # Muniru A Asiru, Mar 04 2018

with(numtheory): seq(sigma(8*n-1)/8, n=1..10^3); # Muniru A Asiru, Mar 04 2018

a[n_] := 8 (DivisorSigma[1, n] - n) + 4; Array[a, 62] (* Robert G. Wilson v, Dec 12 2017 *)

a(n) = 4*A294015(n).
a(n) = 8*(A001065(n) + (1/2)).
a(n) = 8*(A000203(n) - n + (1/2)).
a(n) = A239050(n) - 4*A235796(n).
a(n) = A017113(n-1) - 8*A235796(n).

A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 6, 1, 2, 1, 10, 1, 12, 3, 8, 1, 16, 1, 18, 1, 4, 5, 22, 1, 4, 3, 2, 3, 28, 2, 30, 1, 20, 4, 24, 1, 36, 9, 8, 1, 40, 1, 42, 5, 8, 11, 46, 1, 6, 1, 32, 3, 52, 1, 8, 3, 4, 7, 58, 1, 60, 15, 4, 1, 48, 5, 66, 2, 44, 6, 70, 1, 72, 9, 8, 9, 60, 2, 78

Offset: 1

Amiram Eldar, Jun 01 2024

Numbers that are unitarily divided by n are numbers k such that n is a unitary divisor of k, or equivalently, numbers of the form m*n, with gcd(m, n) = 1.

			Fractions begin with: 1, 1/4, 2/9, 1/8, 4/25, 1/18, 6/49, 1/16, 2/27, 1/25, 10/121, 1/36, ...
For n = 2, the numbers that are unitarily divided by 2 are the numbers of the form 4*k+2 whose asymptotic density is 1/4. Therefore a(2) = numerator(1/4) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A000010, A003277, A034444, A064549, A076512, A077610, A109395, A173557, A373319(denominators), A373320.
Cf. A001620, A013661, A306016.
Numbers that are unitarily divided by k: A000027 (k=1), A016825 (k=2), A016051 (k=3), A017113 (k=4), A051062 (k=8), A051063 (k=9).

a[n_] := Numerator[EulerPhi[n]/n^2]; Array[a, 100]

PARI
```
a(n) = numerator(eulerphi(n)/n^2);
```

a(n) = 1 if and only if n is in A090778.
a(n) = A000010(n) if and only if n is a cyclic number (A003277).
Let f(n) = a(n)/A373319(n). Then:
f(n) = A000010(n)/n^2 = A076512(n)/(n*A109395(n)).
f(n) = A173557(n)/A064549(n).
f(n) is multiplicative with f(p^e) = (1 - 1/p)/p^e.
Sum_{k=1..n} f(k) = (log(n) + gamma - zeta'(2)/zeta(2)) / zeta(2), where gamma is Euler's constant (A001620).

A017114 a(n) = (8*n + 4)^2.

Original entry on oeis.org

16, 144, 400, 784, 1296, 1936, 2704, 3600, 4624, 5776, 7056, 8464, 10000, 11664, 13456, 15376, 17424, 19600, 21904, 24336, 26896, 29584, 32400, 35344, 38416, 41616, 44944, 48400, 51984, 55696, 59536, 63504, 67600, 71824, 76176, 80656, 85264, 90000, 94864, 99856

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. A006752, A016754, A016826, A017113, A061042.

[(8*n+4)^2: n in [0..35] ]; // Vincenzo Librandi, Jul 21 2011

LinearRecurrence[{3, -3, 1},{16, 144, 400},30] (* Ray Chandler, Aug 04 2015 *)
(8*Range[0,40]+4)^2 (* Harvey P. Dale, Aug 13 2024 *)

a(n)=(8*n+4)^2 \\ Charles R Greathouse IV, Jun 17 2017

From Paul Curtz, Nov 07 2008: (Start)
a(n) = 16*A016754(n).
a(n+2) = A061042(2n+1), from Brackett spectrum of hydrogen. (End)
G.f.: -16*(1 + 6*x + x^2)/(x-1)^3. - R. J. Mathar, Jul 14 2016
From Amiram Eldar, Apr 25 2023: (Start)
a(n) = A017113(n)^2.
a(n) = 2^2*A016826(n).
Sum_{n>=0} 1/a(n) = Pi^2/128.
Sum_{n>=0} (-1)^n/a(n) = G/16, where G is Catalan's constant (A006752). (End)

cp's OEIS Frontend

A329066 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( (Sum_{j=0..n} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n} y^(2*j+1)+1/y^(2*j+1)) - (Sum_{j=0..n-1} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n-1} y^(2*j+1)+1/y^(2*j+1)) )^(2*k).

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A047457 Numbers that are congruent to {3, 4} mod 8.

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A374933 Maximum number of squares covered (i.e., attacked) by 2 independent (i.e., non-attacking) queens on an n X n chessboard.

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A337940 Triangle read by rows: T(n, k) = T(n+2) - T(n-k), with the triangular numbers T = A000217, for n >= 1, k = 1, 2, ..., n.

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A365883 Numbers k whose least prime divisor is equal to its exponent in the prime factorization of k.

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Maple

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PARI

A374934 Maximum number of squares covered (i.e., attacked) by 4 independent (i.e., nonattacking) queens on an n X n chessboard.

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A001533 a(n) = (8*n+1)*(8*n+7).

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A294628 a(n) = 8*(sigma(n) - n + (1/2)).

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GAP

A329066 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ( (Sum_{j=0..n} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n} y^(2j+1)+1/y^(2j+1)) - (Sum_{j=0..n-1} x^(2j+1)+1/x^(2j+1)) * (Sum_{j=0..n-1} y^(2j+1)+1/y^(2j+1)) )^(2*k).

A001533 a(n) = (8n+1)(8*n+7).