A210530 -id:A210530 - cp's OEIS frontend

A109043 a(n) = lcm(n,2).

0, 2, 2, 6, 4, 10, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 18, 38, 20, 42, 22, 46, 24, 50, 26, 54, 28, 58, 30, 62, 32, 66, 34, 70, 36, 74, 38, 78, 40, 82, 42, 86, 44, 90, 46, 94, 48, 98, 50, 102, 52, 106, 54, 110, 56, 114, 58, 118, 60, 122, 62, 126, 64, 130, 66, 134

Offset: 0

Mitch Harris, Jun 18 2005

Exponent of the dihedral group D(2n) = . - Arkadiusz Wesolowski, Sep 10 2013
Second column of table A210530. - Boris Putievskiy, Jan 29 2013
For n > 1, the basic period of A000166(k) (mod n) (Miska, 2016). - Amiram Eldar, Mar 03 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, The Exponent of a Group: Properties, Computations and Applications, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
Piotr Miska, Arithmetic properties of the sequence of derangements, Journal of Number Theory, Vol. 163 (2016), pp. 114-145; arXiv preprint, arXiv:1508.01987 [math.NT], 2015. See p. 124 (p. 14 in the preprint).
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).
Index entries for sequences related to lcm's.

Cf. A000166, A109042, A152749 (partial sums).
Cf. A066043 (essentially the same), A000034 (=a(n)/n), A026741 (=a(n)/2).
Cf. A186421, A210530.

a109043 = (lcm 2)
a109043_list = zipWith (*) [0..] a000034_list
-- Reinhard Zumkeller, Mar 31 2012

[0, 2, 2] cat [Exponent(DihedralGroup(n)) : n in [3..65]]; // Arkadiusz Wesolowski, Sep 10 2013

LCM[Range[0,70],2] (* Harvey P. Dale, Aug 19 2012 *)

a(n)=lcm(n,2) \\ Charles R Greathouse IV, Sep 24 2015

def A109043(n): return n<<1 if n&1 else n # Chai Wah Wu, Aug 05 2024

[lcm(n,2) for n in range(0, 68)] # Zerinvary Lajos, Jun 07 2009

For n > 0: a(n) = A186421(n) + A186421(n+2).
a(n) = n*2 / gcd(n, 2).
a(n) = -(n*((-1)^n-3))/2. - Stephen Crowley, Feb 11 2007
From R. J. Mathar, Aug 20 2008: (Start)
a(n) = A066043(n), n > 1.
a(n) = 2*A026741(n).
G.f.: 2*x(1+x+x^2)/((1-x)^2*(1+x)^2). (End)
a(n) = n*A000034(n). - Paul Curtz, Mar 25 2011
E.g.f.: x*(2*cosh(x) + sinh(x)). - Stefano Spezia, May 09 2021
Sum_{k=1..n} a(k) ~ (3/4) * n^2. - Amiram Eldar, Nov 26 2022

A065423 Number of ordered length 2 compositions of n with at least one even summand.

Original entry on oeis.org

0, 0, 2, 1, 4, 2, 6, 3, 8, 4, 10, 5, 12, 6, 14, 7, 16, 8, 18, 9, 20, 10, 22, 11, 24, 12, 26, 13, 28, 14, 30, 15, 32, 16, 34, 17, 36, 18, 38, 19, 40, 20, 42, 21, 44, 22, 46, 23, 48, 24, 50, 25, 52, 26, 54, 27, 56, 28, 58, 29, 60, 30, 62, 31, 64, 32, 66, 33, 68, 34, 70, 35, 72, 36, 74

Offset: 1

Len Smiley, Nov 23 2001

			a(7) = 6 because we can write 7 = 1+6 = 2+5 = 3+4 = 4+3 = 5+2 = 6+1; a(8) = 3 because we can write 8 = 2+6 = 4+4 = 6+2.

Stefano Spezia, Table of n, a(n) for n = 1..10000
Mircea Merca and Emil Simion, n-Color Partitions into Distinct Parts as Sums over Partitions, Symmetry (2023) Vol. 15, Iss. 11.
Index entries for two-way infinite sequences
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A026741, A097140 (first differences), A030451 (absolute first differences), A210530.

int a(int n){n--;return n>>(n&1);} // Mia Boudreau, Aug 27 2025

A065423 := proc(n)
    (3*n-4-(-1)^n*n)/4 ;
end proc:
seq(A065423(n),n=1..40) ; # R. J. Mathar, Jan 24 2022

LinearRecurrence[{0,2,0,-1},{0,0,2,1},100] (* Harvey P. Dale, May 14 2014 *)

PARI
```
a(n)=n-=2;if(n%2,n+1,n/2)
```

G.f.: x^3*(x+2)/(1-x^2)^2.
a(n) = floor((n-1)/2) + (n is odd)*floor((n-1)/2).
a(n+2) = Sum_{k=0..n} (gcd(n, k) mod 2). - Paul Barry, May 02 2005
a(n) = Sum_{i=1..n-1} (-1)^i (floor(i/2) + ((i+1) mod 2)). - Olivier Gérard, Jun 21 2007
a(n) = A210530(n,4)/2 for n>2. - Boris Putievskiy, Jan 29 2013
a(n) = (3*n-4-n*(-1)^n)/4. - Boris Putievskiy, Jan 29 2013, corrected Jan 24 2022
a(n) = A026741(n)-1. - Wesley Ivan Hurt, Jun 23 2013
a(n) = floor((n-1) / 2^mod(n-1,2)). - Mia Boudreau, Aug 27 2025
E.g.f.: 1 + (x - 1)*cosh(x) + (x - 2)*sinh(x)/2. - Stefano Spezia, Dec 17 2023

A114753 First column of A114751.

Original entry on oeis.org

1, 3, 3, 7, 5, 11, 7, 15, 9, 19, 11, 23, 13, 27, 15, 31, 17, 35, 19, 39, 21, 43, 23, 47, 25, 51, 27, 55, 29, 59, 31, 63, 33, 67, 35, 71, 37, 75, 39, 79, 41, 83, 43, 87, 45, 91, 47, 95, 49, 99, 51, 103, 53, 107, 55, 111, 57, 115, 59, 119, 61, 123, 63, 127, 65, 131, 67, 135, 69

Offset: 1

Amarnath Murthy, Nov 15 2005

A114752, (1, 2, 5, 4, 9, 6, ...) + A114753 - (1,1,1,...) = 3n+1: (1, 4, 7, 10, 13, ...). - Gary W. Adamson, Sep 16 2007

First column of table A210530. - Boris Putievskiy, Jan 29 2013

Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A114751, A114752.

Cf. A133080, A210530.

Table[If[OddQ[n],n,2n-1],{n,1,120}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)

a(2n+1) = 2n+1, a(2n) = 4n-1.
a(n) = 2*a(n-2) - a(n-4). - Joerg Arndt, Apr 02 2011
Equals A133080 * [1,2,3,...]. - Gary W. Adamson, Sep 08 2007
G.f. x*(1+3*x+x^2+x^3) / ( (x-1)^2*(1+x)^2 ). - R. J. Mathar, Apr 04 2012
a(n) = (3*n-1-(n-1)*(-1)^(n-1))/2. - Boris Putievskiy, Jan 29 2013

More terms from Joshua Zucker, May 05 2006

A066104 a(2n) = 2n, a(2n+1) = 4(n+1).

Original entry on oeis.org

0, 4, 2, 8, 4, 12, 6, 16, 8, 20, 10, 24, 12, 28, 14, 32, 16, 36, 18, 40, 20, 44, 22, 48, 24, 52, 26, 56, 28, 60, 30, 64, 32, 68, 34, 72, 36, 76, 38, 80, 40, 84, 42, 88, 44, 92, 46, 96, 48, 100, 50, 104, 52, 108, 54, 112, 56, 116, 58, 120, 60, 124, 62, 128, 64, 132, 66, 136

Offset: 0

George E. Antoniou, Dec 04 2001

Fourth column of table A210530 for n>0. - Boris Putievskiy, Jan 29 2013

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A065423, A210530.

Table[(3*n+2-(n+2)*(-1)^n)/2, {n,0,50}] (* or *) LinearRecurrence[{0, 2, 0, -1}, {0, 4, 2, 8}, 50] (* G. C. Greubel, Dec 24 2016 *)

{ for (n=0, 1000, if(n%2, a=2*n + 2, a=n); write("b066104.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 14 2009

concat([0], Vec(2*x*(x+2)/(1-x^2)^2 + O(x^50))) \\ G. C. Greubel, Dec 24 2016

a(n) = 2*A065423(n+1).
O.g.f.: 2*x(2+x)/(1-x^2)^2. - Len Smiley, Dec 06 2001
a(n) = (3*n+2-(n+2)*(-1)^n)/2. - Boris Putievskiy, Jan 29 2013

A114751 The following triangle contains n consecutive numbers beginning from n in ascending order if n is odd else in descending order. Sequence contains the triangle by rows.

Original entry on oeis.org

1, 3, 2, 3, 4, 5, 7, 6, 5, 4, 5, 6, 7, 8, 9, 11, 10, 9, 8, 7, 6, 7, 8, 9, 10, 11, 12, 13, 15, 14, 13, 12, 11, 10, 9, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12

Offset: 1

Amarnath Murthy, Nov 15 2005

			1
3 2
3 4 5
7 6 5 4
5 6 7 8 9
11 10 9 8 7 6
...

Cf. A114752, A114753, A210530.

for n from 1 to 14 do if n mod 2 = 1 then print(seq(k,k=n..2*n-1)) else print(seq(2*n-k,k=1..n)) fi od; # yields sequence in triangular form # Emeric Deutsch, Jan 26 2006

a(n) = (3*t+2-t*(-1)^(t-1))/2-(1+(-1)^t)*(j-1)/2+(1-(-1)^t)*(j-1)/2, where j = (t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Jan 30 2013

More terms from Emeric Deutsch, Jan 26 2006

A211161 Table T(n,k) = n, if k is odd, k/2 if k is even; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 2, 1, 4, 1, 2, 3, 1, 5, 3, 2, 2, 4, 1, 6, 1, 3, 3, 2, 5, 1, 7, 4, 2, 3, 4, 2, 6, 1, 8, 1, 4, 3, 3, 5, 2, 7, 1, 9, 5, 2, 4, 4, 3, 6, 2, 8, 1, 10, 1, 5, 3, 4, 5, 3, 7, 2, 9, 1, 11, 6, 2, 5, 4, 4, 6, 3, 8, 2, 10, 1, 12, 1, 6, 3, 5, 5, 4, 7, 3

Offset: 1

Boris Putievskiy, Jan 30 2013

In general, let B and C be sequences. By b(n) and c(n)denote elements B and C respectively. Table T(n,k) = b(n), if k is odd, c(k) if k is even. For this sequence b(n)=n, c(k)=k.

Row T(n,k) is b(n),c(1),b(n),c(2),b(n),c(3),...Numbers c(1),c(2),c(3),... sandwiched between b(n)'s. For this sequence numbers 1,2,3,... (A000027) sandwiched between n's.

			The start of the sequence as table for general case:
b(1)..c(1)..b(1)..c(2)..b(1)..c(3)..b(1)..c(4)...
b(2)..c(1)..b(2)..c(2)..b(2)..c(3)..b(2)..c(4)...
b(3)..c(1)..b(3)..c(2)..b(3)..c(3)..b(3)..c(4)...
b(4)..c(1)..b(4)..c(2)..b(4)..c(3)..b(4)..c(4)...
b(5)..c(1)..b(5)..c(2)..b(5)..c(3)..b(5)..c(4)...
b(6)..c(1)..b(6)..c(2)..b(6)..c(3)..b(6)..c(4)...
b(7)..c(1)..b(7)..c(2)..b(7)..c(3)..b(7)..c(4)...
b(8)..c(1)..b(8)..c(2)..b(8)..c(3)..b(8)..c(4)...
. . .
The start of the sequence as triangle array read by rows for general case:
b(1);
c(1),b(2);
b(1),c(1),b(3);
c(2),b(2),c(1),b(4);
b(1),c(2),b(3),c(1),b(5);
c(3),b(2),c(2),b(4),c(1),b(6);
b(1),c(3),b(3),c(2),b(5),c(1),b(7);
c(4),b(2),c(3),b(4),c(2),b(6),c(1),b(8);
. . .
Row number r contains r numbers.
If r is odd  b(1),c((r-1)/2),b(3),c((r-1)/2-1),b(5),c((r-1)/2-2),...c(1),b(r).
If r is even c(r/2),b(2),c(r/2-1),b(4),c(r/2-2),b(6),...c(1),b(r).
The start of the sequence as table for b(n)=n and c(k)=k:
1..1..1..2..1..3..1..4...
2..1..2..2..2..3..2..4...
3..1..3..2..3..3..3..4...
4..1..4..2..4..3..4..4...
5..1..5..2..5..3..5..4...
6..1..6..2..6..3..6..4...
7..1..7..2..7..3..7..4...
8..1..8..2..8..3..8..4...
. . .
The start of the sequence as triangle array read by rows for b(n)=n and c(k)=k:
1;
1,2;
1,1,3;
2,2,1,4;
1,2,3,1,5;
3,2,2,4,1,6;
1,3,3,2,5,1,7;
4,2,3,4,2,6,1,8;
. . .
Row number r contains r numbers.
If r is odd  1,(r-1)/2,3,(r-1)/2-1,5,(r-1)/2-2,...1,r.
If r id even r/2,2,r/2-1,4,r/2-1,6,...1,r.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A000027, A002260, A004736, A210530.

def a(n):
    t=int((math.sqrt(8*n-7) - 1)/ 2)
    i=n-t*(t+1)//2
    j=(t*t+3*t+4)//2-n
    return (1+(-1)**j)*j//4 - (-1+(-1)**j)*i//2

For the general case
As table T(n,k) = (1+(-1)^k)*c(k/2)/2 - (-1+(-1)^k)*b(n)/2.
As linear sequence
a(n) = (1+(-1)^j)*c(j/2)/2 - (-1+(-1)^j)*b(i)/2, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).
For b(n) = n and c(k) = k:
As table T(n,k) = (1+(-1)^k)*k/4 - (-1+(-1)^k)*n/2.
As linear sequence a(n) = (1+(-1)^A004736(n))*A004736(n)/4 - (-1+(-1)^A004736(n))*A002260(n)/2. a(n) = (1+(-1)^j)*j/4 - (-1+(-1)^j)*i/2,
where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).

A204904 p(n)-q(n), where (p(n), q(n)) is the least pair of odd primes for which n divides p(n)-q(n).

Original entry on oeis.org

2, 2, 6, 4, 10, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 18, 38, 20, 42, 22, 46, 24, 50, 26, 54, 28, 58, 30, 62, 32, 66, 34, 70, 36, 74, 38, 78, 40, 82, 42, 86, 44, 90, 46, 94, 48, 98, 50, 102, 52, 106, 54, 110, 56, 114, 58, 118, 60, 122, 62, 126, 64, 130

Offset: 1

Clark Kimberling, Jan 20 2012

nonn

This sequence agrees with A109043 for 0

For a guide to related sequences, see A204892.

Sequence agrees with A109043 at least up to 6400. - Michel Marcus, Mar 14 2018

If Polignac's conjecture is true, then this is a duplicate of A109043. - Robert Israel, Mar 14 2018

			1 = (5-3)/2=(7-3)/4=(13-3)/6=(11-3)/8=...
2 = (5-3)/1=(11-5)/3=(7-3)/5=(17-3)/7=...

Wikipedia, Polignac's conjecture

Cf. A066043, A109043, A204900, A204892, A210530.

Mathematica
```
(See the program at A204900.)
```

a(n) = {forprime(p=5,,forprime(q=3, p-1, d = p-q; if ((d % n) == 0, return (d));););} \\ Michel Marcus, Mar 14 2018

A211197 Table T(n,k) = 2n + ((-1)^n)(1/2 - (k-1) mod 2) - 1/2; n, k > 0, read by antidiagonals.

Original entry on oeis.org

1, 2, 4, 1, 3, 5, 2, 4, 6, 8, 1, 3, 5, 7, 9, 2, 4, 6, 8, 10, 12, 1, 3, 5, 7, 9, 11, 13, 2, 4, 6, 8, 10, 12, 14, 16, 1, 3, 5, 7, 9, 11, 13, 15, 17, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22

Offset: 1

Boris Putievskiy, Feb 03 2013

In general, let B and C be sequences. By b(n) and c(n) denote elements B and C respectively. Table T(n,k) = (1-(-1)^k)*b(n)/2+(1+(-1)^k)*c(n)/2 read by antidiagonals.
For this sequence b(n)=2*n-1, b(n)=A005408(n), c(n)=2*n, c(n)=A005843(n).
If n is odd   row T(n,k) is alternation b(n) and c(n) starts from b(n).
If n is even  row T(n,k) is alternation c(n) and b(n) starts from c(n).
For this sequence if n is odd   alternation numbers  2*n-1 and 2*n   starts from  2*n-1.
For this sequence if n is even  alternation numbers  2*n   and 2*n-1 starts from  2*n.
T(n,k) is replication of the first and the second columns that are “a braid” from sequences B and C.

			The start of the sequence as table for general case:
  b(1)..c(1)..b(1)..c(1)..b(1)..c(1)..b(1)..c(1)..
  c(2)..b(2)..c(2)..b(2)..c(2)..b(2)..c(2)..b(2)..
  b(3)..c(3)..b(3)..c(3)..b(3)..c(3)..b(3)..c(3)..
  c(4)..b(4)..c(4)..b(4)..c(4)..b(4)..c(4)..b(4)..
  b(5)..c(5)..b(5)..c(5)..b(5)..c(5)..b(5)..c(5)..
  c(6)..b(6)..c(6)..b(6)..c(6)..b(6)..c(6)..b(6)..
  b(7)..c(7)..b(7)..c(7)..b(7)..c(7)..b(7)..c(7)..
  c(8)..b(8)..c(8)..b(8)..c(8)..b(8)..c(8)..b(8)..
  . . .
The start of the sequence as triangle array read by rows for general case:
  b(1);
  c(1),c(2);
  b(1),b(2),b(3);
  c(1),c(2),c(3),c(4);
  b(1),b(2),b(3),b(4),b(5);
  c(1),c(2),c(3),c(4),c(5),c(6);
  b(1),b(2),b(3),b(4),b(5),b(6),b(7);
  c(1),c(2),c(3),c(4),c(5),c(6),c(7),c(8);
. . .
Row number r contains r numbers.
If r is odd  b(1),b(2),...,b(r).
If r is even c(1),c(2),...,c(r).
The start of the sequence as table for b(n)=2*n-1 and c(n)=2*n:
  1....2...1...2...1...2...1...2...
  4....3...4...3...4...3...4...3...
  5....6...5...6...5...6...5...6...
  8....7...8...7...8...7...8...7...
  9...10...9..10...9..10...9..10...
  12..11..12..11..12..11..12..11...
  13..14..13..14..13..14..13..14...
  16..15..16..15..16..15..16..15...
  . . .
The start of the sequence as triangle array read by rows for  b(n)=2*n-1 and c(n)=2*n:
  1;
  2,4;
  1,3,5;
  2,4,6,8;
  1,3,5,7,9;
  2,4,6,8,10,12;
  1,3,5,7,9,11,13;
  2,4,6,8,10,12,14,16;
  . . .
Row number r contains r numbers.
If r is odd  1,3,...2*r-1 - coincides with the elements row number r triangle array read by rows for sequence 2*A002260-1.
If r is even 2,4,...,2*r  - coincides with the elements row number r triangle array read by rows for sequence 2*A002260.

Boris Putievskiy, Rows n = 1..140 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A000027, A003056, A002260, A004736, A210530, A158405, A005408, A005843, A057211.

t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2
j=(t*t+3*t+4)/2-n
result =2*i+((-1)**i)*(0.5 - (j-1) % 2) - 0.5

a211197_list = [2*n - k%2 for k in range(1, 13) for n in range(1, k+1)] # David Radcliffe, Jun 01 2025

For the general case:
As a table: T(n,k) = (1-(-1)^k)*b(n)/2+(1+(-1)^k)*c(n)/2.
As a linear sequence: a(n) = (1-(-1)^j)*b(i)/2+(1+(-1)^j)*c(i)/2, where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).
where b(n) = 2*n-1 and c(n) = 2*n.
As a table: T(n,k) = 2*n+((-1)^n)*(1/2- (k-1) mod 2) - 1/2.
As a linear sequence:
a(n) = 2*A002260(n) + ((-1)^A002260(n))*(1/2- (A004736(n)-1) mod 2) -1/2.
a(n) = -(1+(-1)^A003056(n))*A002260(n) +(1+(-1)^A003056(n))*(2*A002260(n)-1)/2.
a(n) =  2*i+((-1)^i)*(1/2- (j-1) mod 2) - 1/2
a(n) = -(1+(-1)^t)*i +(1+(-1)^t)*(2*i-1)/2,
where i = n-t*(t+1)/2, j = (t*t+3*t+4)/2-n, t = floor((-1+sqrt(8*n-7))/2).
T(n, k) = 2*n - 1 + (n+k mod 2); a(n) = 2*A002260(n) - A057211(n). - David Radcliffe, Jun 01 2025

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cp's OEIS Frontend

A109043 a(n) = lcm(n,2).

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A065423 Number of ordered length 2 compositions of n with at least one even summand.

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A114753 First column of A114751.

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A066104 a(2n) = 2n, a(2n+1) = 4(n+1).

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PARI

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A114751 The following triangle contains n consecutive numbers beginning from n in ascending order if n is odd else in descending order. Sequence contains the triangle by rows.

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Maple

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Extensions

A211161 Table T(n,k) = n, if k is odd, k/2 if k is even; n, k > 0, read by antidiagonals.

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Python

Formula

A204904 p(n)-q(n), where (p(n), q(n)) is the least pair of odd primes for which n divides p(n)-q(n).

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A211197 Table T(n,k) = 2n + ((-1)^n)(1/2 - (k-1) mod 2) - 1/2; n, k > 0, read by antidiagonals.