A109043 -id:A109043 - cp's OEIS frontend

A168077 a(2n) = A129194(2n)/2; a(2n+1) = A129194(2n+1).

0, 1, 1, 9, 4, 25, 9, 49, 16, 81, 25, 121, 36, 169, 49, 225, 64, 289, 81, 361, 100, 441, 121, 529, 144, 625, 169, 729, 196, 841, 225, 961, 256, 1089, 289, 1225, 324, 1369, 361, 1521, 400, 1681, 441, 1849, 484, 2025, 529, 2209, 576, 2401, 625, 2601

Offset: 0

Paul Curtz, Nov 18 2009

From Paul Curtz, Mar 26 2011: (Start)
Successive A026741(n) * A026741(n+p):
  p=0:  0, 1,  1,  9,  4, 25,  9,     a(n),
  p=1:  0, 1,  3,  6, 10, 15, 21,  A000217,
  p=2:  0, 3,  2, 15,  6, 35, 12,  A142705,
  p=3:  0, 2,  5,  9, 14, 20, 27,  A000096,
  p=4:  0, 5,  3, 21,  8, 45, 15,  A171621,
  p=5:  0, 3,  7, 12, 18, 25, 33,  A055998,
  p=6:  0, 7,  4, 27, 10, 55, 18,
  p=7:  0, 4,  9, 15, 22, 30, 39,  A055999,
  p=8:  0, 9,  5, 33, 12, 65, 21,  (see A061041),
  p=9:  0, 5, 11, 18, 26, 35, 45,  A056000. (End)
The moment generating function of p(x, m=2, n=1, mu=2) = 4*x*E(x, 2, 1), see A163931 and A274181, is given by M(a) = (-4 * log(1-a) - 4 * polylog(2, a))/a^2. The series expansion of M(a) leads to the sequence given above. - Johannes W. Meijer, Jul 03 2016
Multiplicative because both A129194 and A040001 are. - Andrew Howroyd, Jul 26 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A040001, A168068, A181318.

I:=[0,1,1,9,4,25]; [n le 6 select I[n] else 3*Self(n-2)-3*Self(n-4)+Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jul 10 2016

a := proc(n): n^2*(5-3*(-1)^n)/8 end: seq(a(n), n=0..46); # Johannes W. Meijer, Jul 03 2016

LinearRecurrence[{0,3,0,-3,0,1},{0,1,1,9,4,25},60] (* Harvey P. Dale, May 14 2011 *)
f[n_] := Numerator[(n/2)^2]; Array[f, 60, 0] (* Robert G. Wilson v, Dec 18 2012 *)
CoefficientList[Series[x(1+x+6x^2+x^3+x^4)/((1-x)^3(1+x)^3), {x,0,60}], x] (* Vincenzo Librandi, Jul 10 2016 *)

concat(0, Vec(x*(1+x+6*x^2+x^3+x^4)/((1-x)^3*(1+x)^3) + O(x^60))) \\ Altug Alkan, Jul 04 2016

a(n) = lcm(4, n^2)/4; \\ Andrew Howroyd, Jul 26 2018

(x*(1+x+6*x^2+x^3+x^4)/(1-x^2)^3).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, Feb 20 2019

From R. J. Mathar, Jan 22 2011: (Start)
G.f.: x*(1 + x + 6*x^2 + x^3 + x^4) / ((1-x)^3*(1+x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).
a(n) = n^2*(5 - 3*(-1)^n)/8. (End)
a(n) = A026741(n)^2.
a(2*n) = A000290(n); a(2*n+1) = A016754(n).
a(n) - a(n-4) = 4*A064680(n+2). - Paul Curtz, Mar 27 2011
4*a(n) = A061038(n) * A010121(n+2) = A109043(n)^2, n >= 2. - Paul Curtz, Apr 07 2011
a(n) = A129194(n) / A040001(n). - Andrew Howroyd, Jul 26 2018
From Peter Bala, Feb 19 2019: (Start)
a(n) = numerator(n^2/(n^2 + 4)) = n^2/(gcd(n^2,4)) = (n/gcd(n,2))^2.
a(n) = n^2/b(n), where b(n) = [1, 4, 1, 4, ...] is a purely periodic sequence of period 2. Thus a(n) is a quasi-polynomial in n.
O.g.f.: x*(1 + x)/(1 - x)^3 - 3*x^2*(1 + x^2)/(1 - x^2)^3.
Cf. A181318. (End)
From Werner Schulte, Aug 30 2020: (Start)
Multiplicative with a(2^e) = 2^(2*e-2) for e > 0, and a(p^e) = p^(2*e) for prime p > 2.
Dirichlet g.f.: zeta(s-2) * (1 - 3/2^s).
Dirichlet convolution with A259346 equals A000290.
Sum_{n>0} 1/a(n) = Pi^2 * 7 / 24. (End)
Sum_{k=1..n} a(k) ~ (5/24) * n^3. - Amiram Eldar, Nov 28 2022

A210530 T(n,k) = (k + 3n - 2 - (k+n-2)(-1)^(k+n))/2 n, k > 0, read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Jan 28 2013

Row T(n,k) for odd n is even numbers sandwiched between n's starts from n and 2*n.
Row T(n,k) for even n is odd numbers sandwiched between n's starts from 2*n-1 and n.
Antidiagonal T(1,k), T(2,k-1), ..., T(k,1) for odd k is 1,2,3,...,k.
Antidiagonal T(1,k), T(2,k-1), ..., T(k,1) for even k is k+1, k+2, ..., 2*k+1.
The main diagonal is A000027.
Diagonal, located above the main diagonal T(1,k), T(2,k+1), T(3,k+2), ... for odd k is A000027.
Diagonal, located above the main diagonal T(1,k), T(2,k+1), T(3,k+2), ... for even k is k, k+3, k+6, ..., A016789, A016777, A008585.
Diagonal, located below the main diagonal T(n,1), T(n+1,2), T(n+2,3), ... for odd n is n,n+1, n+2, ... A000027.
Diagonal, located below the main diagonal T(n,1), T(n+1,2), T(n+2,3), ... for even n is 2*n-1, 2*n+2, 2*n+5, ... A008585, A016777, A016789.
The table contains:
A124625 as row 1,
A114753 as column 1,
A109043 as column 2,
A066104 as column 4.

			The start of the sequence as table:
   1   2   1   4   1   6   1   8   1  10
   3   2   5   2   7   2   9   2  11   2
   3   6   3   8   3  10   3  12   3  14
   7   4   9   4  11   4  13   4  15   4
   5  10   5  12   5  14   5  16   5  18
  11   6  13   6  15   6  17   6  19   6
   7  14   7  16   7  18   7  20   7  22
  15   8  17   8  19   8  21   8  23   8
   9  18   9  20   9  22   9  24   9  26
  19  10  21  10  23  10  25  10  27  10
  ...
The start of the sequence as triangle array read by rows:
   1;
   2,  3;
   1,  2,  3;
   4,  5,  6,  7;
   1,  2,  3,  4,  5;
   6,  7,  8,  9, 10, 11;
   1,  2,  3,  4,  5,  6,  7;
   8,  9, 10, 11, 12, 13, 14, 15;
   1,  2,  3,  4,  5,  6,  7,  8,  9;
  10, 11, 12, 13, 14, 15, 16, 17, 18, 19;
  ...
Row number r contains r numbers.
If r is  odd: 1,2,3,...,r.
If r is even: r, r+1, r+3, ..., 2*r-1.
The start of the sequence as array read by rows, the length of row r is 4*r-1.
First 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.
Last 2*r numbers are from the row number 2*r of triangle array, located above.
  1,2,3;
  1,2,3,4,5,6,7;
  1,2,3,4,5,6,7,8,9,10,11;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;
  1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19;
  ...
Row number r contains 4*r-1 numbers: 1,2,3,...,4*r-1.

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. A124625, A114753, A109043, A066104, A000027, A016789, A016777, A008585, A204164.

T[n_, k_] := (k+3n-2-(k+n-2)(-1)^(k+n))/2; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

T(n,k) = (k+3*n-2-(k+n-2)*(-1)^(k+n))/2; \\ Andrew Howroyd, Jan 11 2018

t=int((math.sqrt(8*n-7)-1)/2)
v=int((t+2)/2)
result=n-v*(2*v-3)-1

As table T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2.
As linear sequence
a(n) = A000027(n) - A204164(n)*(2*A204164(n)-3) - 1.
a(n) = n - v*(2*v-3) - 1, where t = floor((-1 + sqrt(8*n-7))/2) and v = floor((t+2)/2).
G.f. of the table: (y*(- 1 + 3*y^2) + x^2*(2 + 5*y - 2*y^2 - 7*y^3) + x^3*(4 + y - 6*y^2 - y^3) + x*(y + 2*y^2 - y^3))/((- 1 + x)^2*(1 + x)^2*(-1 + y)^2*(1 + y)^2). - Stefano Spezia, Nov 17 2018

A186421 Even numbers interleaved with repeated odd numbers.

Original entry on oeis.org

0, 1, 2, 1, 4, 3, 6, 3, 8, 5, 10, 5, 12, 7, 14, 7, 16, 9, 18, 9, 20, 11, 22, 11, 24, 13, 26, 13, 28, 15, 30, 15, 32, 17, 34, 17, 36, 19, 38, 19, 40, 21, 42, 21, 44, 23, 46, 23, 48, 25, 50, 25, 52, 27, 54, 27, 56, 29, 58, 29, 60, 31, 62, 31, 64, 33, 66, 33, 68, 35, 70, 35, 72, 37, 74, 37, 76, 39, 78, 39, 80, 41, 82, 41, 84, 43, 86, 43

Offset: 0

Reinhard Zumkeller, Feb 21 2011

A005843 interleaved with A109613.

Row sum of superimposed binary filled triangle. - Craig Knecht, Aug 07 2015

			A005843: 0   2   4   6   8   10   12   14   16   18   20    22
A109613:   1   1   3   3   5    5    7    7    9    9    11    11
this   : 0 1 2 1 4 3 6 3 8 5 10 5 12 7 14 7 16 9 18 9 20 11 22 ... .

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Craig Knecht, Row sums of superimposed binary triangles.
Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Cf. A005843, A109043, A109613.

Cf. A186422 (first differences), A186423 (partial sums), A240828 (row sum of superimposed binary triangle).

a186421 n = a186421_list !! n
a186421_list = interleave [0,2..] $ rep [1,3..] where
   interleave (x:xs) ys = x : interleave ys xs
   rep (x:xs) = x : x : rep xs

[IsEven(n) select n else 2*Floor(n/4)+1: n in [0..100]]; // Vincenzo Librandi, Jul 13 2015

A186421:=n->n-(1-(-1)^n)*(n+(-1)^(n*(n+1)/2))/4: seq(A186421(n), n=0..100); # Wesley Ivan Hurt, Aug 07 2015

Table[n - (1 - (-1)^n)*(n + I^(n (n + 1)))/4, {n, 0, 87}] (* or *)
CoefficientList[Series[x (1 + 2 x + 2 x^3 + x^4)/((1 + x^2) (x - 1)^2 (1 + x)^2), {x, 0, 87}], x] (* or *)
n = 88; Riffle[Range[0, n, 2], Flatten@ Transpose[{Range[1, n, 2], Range[1, n, 2]}]] (* Michael De Vlieger, Jul 14 2015 *)

makelist(n-(1-(-1)^n)*(n+%i^(n*(n+1)))/4, n, 0, 90); /* Bruno Berselli, Mar 04 2013 */

def A186421(n): return (n>>1)|1 if n&1 else n # Chai Wah Wu, Jan 31 2023

a(2*k) = 2*k, a(4*k+1) = a(4*k+3) = 2*k+1.
a(n) = n if n is even, else 2*floor(n/4)+1.
a(2*n-(2*k+1)) + a(2*n+2*k+1) = 2*n, 0 <= k < n.
a(n+2) = A109043(n) - a(n).
G.f.: x*(1+2*x+2*x^3+x^4) / ( (1+x^2)*(x-1)^2*(1+x)^2 ). - R. J. Mathar, Feb 23 2011
a(n) = n-(1-(-1)^n)*(n+i^(n(n+1)))/4, where i=sqrt(-1). - Bruno Berselli, Feb 23 2011
a(n) = a(n-2)+a(n-4)-a(n-6) for n>5. - Wesley Ivan Hurt, Aug 07 2015
E.g.f.: (x*cosh(x) + sin(x) + 2*x*sinh(x))/2. - Stefano Spezia, May 09 2021

Edited by Bruno Berselli, Mar 04 2013

A066043 a(1) = 1; for m > 0, a(2m) = 2m, a(2m+1) = 4m+2.

Original entry on oeis.org

1, 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, 68

Offset: 1

George E. Antoniou, Nov 30 2001

Length of period of sequences r(k,n) = floor(sinh(1)*k!) - n*floor(sinh(1)*k!/n) when n is fixed. - Benoit Cloitre, Jun 22 2003

			r(k,7) is sequence 1, 2, 0, 0, 1, 6, 1, 1, 3, 2, 2, 3, 5, 0, 1, 2, 0, 0, 1, 6, 1, 1, 3, 2, 2, 3, 5, 0.... which is periodic with period (1, 2, 0, 0, 1, 6, 1, 1, 3, 2, 2, 3, 5, 0) of length 14 = a(7).

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

Join[{1}, LCM[Range[2, 100], 2]] (* Paolo Xausa, Feb 19 2024 *)

PARI
```
a(n)=if(n<2,1,if(n%2,2*n,n))
```

{ for (n=1, 1000, a=if (n>1 && n%2, 2*n, n); write("b066043.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 08 2009

O.g.f.: (x+2x^2+4x^3-x^5)/(1-x^2)^2. - Len Smiley, Dec 05 2001
a(n)*a(n+3) = -4 + a(n+1)*a(n+2).
From Harry J. Smith, Nov 08 2009: (Start)
a(n) = A109043(n), n > 1.
a(n) = 2*A026741(n), n > 1. (End)

A227316 a(n) = n(n+1) if n == 0 or 1 (mod 4), otherwise a(n) = n(n+1)/2.

Original entry on oeis.org

0, 2, 3, 6, 20, 30, 21, 28, 72, 90, 55, 66, 156, 182, 105, 120, 272, 306, 171, 190, 420, 462, 253, 276, 600, 650, 351, 378, 812, 870, 465, 496, 1056, 1122, 595, 630, 1332, 1406, 741, 780, 1640, 1722, 903, 946, 1980, 2070, 1081, 1128

Offset: 0

Paul Curtz, Jul 06 2013

			a(0) = 2*0 = 0, a(1) = 2*1 = 2, a(2) = 1*3 = 3, a(3) = 1*6 = 6, a(4) = 2*10 = 20.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A000217, A002378, A130658, A169642 (first bisection), A176743, A109043, A227380.

[(3+(-1)^Floor(n/2))*n*(n+1)/4: n in [0..50]]; // Bruno Berselli, Jul 10 2013

a[n_] := n*(n+1)/4*GCD[n-1, 4]*GCD[n, 4]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 10 2013 *)
Table[If[Mod[n,4]<2,n(n+1),(n(n+1))/2],{n,0,50}] (* or *) LinearRecurrence[ {3,-6,10,-12,12,-10,6,-3,1},{0,2,3,6,20,30,21,28,72},50] (* Harvey P. Dale, Aug 26 2016 *)

a(n) = A130658(n+2)*A000217(n), a(-n-1) = A130658(n)*A000217(n).
a(2n) = A169642(n), a(2n+1) = 2*(2*n+1)*A026741(n+1).
a(n) = A176743(n-2)*A176743(n-1).
a(n) = A177002(n+2)*A064038(n+1).
a(n) = 3*a(n-1) -6*a(n-2) +10*a(n-3) -12*a(n-4) +12*a(n-5) -10*(n-6) +6*(n-7) -3*a(n-8) +a(n-9) = 3*a(n-4) -3*a(n-8) +a(n-12).
G.f.: x*(2-3*x+9*x^2+3*x^5+x^6)/((1-x)^3*(1+x^2)^3). - Bruno Berselli, Jul 10 2013
a(n) = (3+(-1)^floor(n/2))*n*(n+1)/4. - Bruno Berselli, Jul 10 2013
Sum_{n>=1} 1/a(n) = 1 + log(2)/2. - Amiram Eldar, Aug 12 2022

A272104 Sum of the even numbers among the larger parts of the partitions of n into two parts.

Original entry on oeis.org

0, 0, 0, 2, 2, 4, 4, 10, 10, 14, 14, 24, 24, 30, 30, 44, 44, 52, 52, 70, 70, 80, 80, 102, 102, 114, 114, 140, 140, 154, 154, 184, 184, 200, 200, 234, 234, 252, 252, 290, 290, 310, 310, 352, 352, 374, 374, 420, 420, 444, 444, 494, 494, 520, 520, 574, 574, 602

Offset: 0

Wesley Ivan Hurt, Apr 20 2016

Essentially, repeated values of A152749.

Sum of the lengths of the distinct rectangles with even length and integer width such that L + W = n, W <= L. For example, a(10) = 14; the rectangles are 2 X 8 and 4 X 6, so 8 + 6 = 14. - Wesley Ivan Hurt, Nov 04 2017

			a(5) = 4; the partitions of 5 into 2 parts are (4,1),(3,2) and the sum of the larger even parts is 4.
a(6) = 4; the partitions of 6 into 2 parts are (5,1),(4,2),(3,3) and the sum of the larger even parts is also 4.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,0,0,2,-2,0,0,-1,1).

Cf. A000326, A001318, A026741, A049450, A049452, A109043, A126964, A152749, A268351, A272212.

[(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4)/2)/8 : n in [0..50]];

A272104:=n->(1+3*(2*n-3-(-1)^n)/2+3*(2*n-3-(-1)^n)^2/8+(2*n-1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4)/2)/8: seq(A272104(n), n=0..100);

Table[(1 + 3(2n-3-(-1)^n)/2 + 3(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8, {n, 0, 50}]
Table[Total@ Map[First, IntegerPartitions[n, {2}] /. {k_, } /; OddQ@ k -> Nothing], {n, 0, 57}] (* _Michael De Vlieger, Apr 20 2016, Version 10.2 *)

concat(vector(3), Vec(2*x^3*(1-x+x^2)*(1+x+x^2)/((1-x)^3*(1+x)^2*(1+x^2)^2) + O(x^50))) \\ Colin Barker, Apr 20 2016

a(n) = (1 + 3*(2n-3-(-1)^n)/2 + 3*(2n-3-(-1)^n)^2/8 + (2n-1-(-1)^n) * (-1)^((2n+1-(-1)^n)/4)/2) / 8.
a(n) = Sum_{i=ceiling(n/2)..n-1} i * (i+1 mod 2).
a(n) = Sum_{i=1..floor(n/2)} (n-i) * (n-i+1 mod 2).
a(2n+1) = a(2n+2) = A152749(n) = 2*A001318(n).
G.f.: 2*x^3*(1-x+x^2)*(1+x+x^2) / ((1-x)^3*(1+x)^2*(1+x^2)^2). - Colin Barker, Apr 20 2016
From Wesley Ivan Hurt, Apr 22 2016, Apr 23 2016: (Start)
a(2n+2)-a(2n) = A109043(n) = 2*A026741(n).
a(4n) = A049450(n) = 2*A000326(n),
a(8n) = A126964(n) = 2*A049452(n),
a(12n) = 2*A268351(n).
a(n+1) = A001318(n) - A272212(n+1). (End)
E.g.f.: ((2 + 3*x*(1 + x))*cosh(x) - 2*(cos(x) + x*cos(x) + x*sin(x)) + (-1 + 3*(-1 + x)*x)*sinh(x))/16. - Ilya Gutkovskiy, Apr 29 2016

A060992 a(n) = Sum_{gcd(i,j) | 0 < i <= j < n and i+j = n}.

Original entry on oeis.org

0, 1, 1, 3, 2, 6, 3, 8, 6, 11, 5, 17, 6, 16, 15, 20, 8, 27, 9, 31, 22, 26, 11, 44, 20, 31, 27, 45, 14, 60, 15, 48, 36, 41, 41, 75, 18, 46, 43, 80, 20, 87, 21, 73, 72, 56, 23, 108, 42, 85, 57, 87, 26, 108, 67, 116, 64, 71, 29, 165, 30, 76

Offset: 1

Reinhard Zumkeller, Feb 14 2002

			a(12) = gcd(1,11) + gcd(2,10) + gcd(3,9) + gcd(4,8) + gcd(5,7) + gcd(6,6) = 1 + 2 + 3 + 4 + 1 + 6 = 17;
a(13) = gcd(1,12) + gcd(2,11) + gcd(3,10) + gcd(4,9) + gcd(5,8) + gcd(6,7) = 1 + 1 + 1 + 1 + 1 + 1 = 6.

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

Cf. A234307, A018804, A109043.

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
for d from 1 to N do
  c:= floor(d/2);
  for n from d to N by d do
    A[n]:= A[n]+c*numtheory:-phi(n/d)
  od
od:
seq(A[i],i=1..N); # Robert Israel, May 11 2018

Table[Sum[GCD[n - i, i], {i, Floor[n/2]}], {n, 100}] (* Wesley Ivan Hurt, Nov 12 2017 *)

a(n) = sumdiv(n, d, (d\2)*eulerphi(n/d)); \\ Michel Marcus, May 11 2018

a(n) = Sum_{d divides n} floor(d/2)*phi(n/d). a(p) = (p-1)/2 for an odd prime p. - Vladeta Jovovic, Dec 21 2004
a(n) = Sum_{i=1..floor(n/2)} gcd(n-i,i). - Wesley Ivan Hurt, Nov 12 2017
G.f.: Sum_{k>=1} phi(k)*x^(2*k)/((1 + x^k)*(1 - x^k)^2). - Ilya Gutkovskiy, Oct 24 2018
a(n) = (A018804(n) - A109043(n))/2. - Ridouane Oudra, May 31 2025

A137933 Least common multiple of n^2 and 2.

Original entry on oeis.org

2, 4, 18, 16, 50, 36, 98, 64, 162, 100, 242, 144, 338, 196, 450, 256, 578, 324, 722, 400, 882, 484, 1058, 576, 1250, 676, 1458, 784, 1682, 900, 1922, 1024, 2178, 1156, 2450, 1296, 2738, 1444, 3042, 1600, 3362, 1764, 3698, 1936, 4050, 2116, 4418, 2304, 4802, 2500, 5202, 2704

Offset: 1

William A. Tedeschi, Feb 29 2008

nonn

Cf. A016742, A077591, A109043, A245058.

LCM[Range[60]^2,2] (* Harvey P. Dale, Jun 06 2018 *)

a(n) = lcm(n^2, 2); \\ Michel Marcus, Mar 13 2018

a(n) = lcm(n^2, 2).
From R. J. Mathar, Mar 06 2008: (Start)
O.g.f.: -2x(1 + 6x^2 + x^4 + 2x^3 + 2x)/((-1+x)^3 * (x+1)^3).
a(2n) = A016742(n).
a(2n+1) = A077591(n). (End)
a(n) = n*A109043(n). - Michel Marcus, Mar 13 2018
From Amiram Eldar, Jul 06 2022: (Start)
Sum_{n>=1} 1/a(n) = 5*Pi^2/48.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/48 = -A245058. (End)

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

A118563 The set of numbers of the form A003173(n)+1 with odd A003173(n) or of the form 2*(A003173(n)+1) with even A003173(n).

Original entry on oeis.org

2, 4, 6, 8, 12, 20, 44, 68, 164

Offset: 1

Roger L. Bagula, May 03 2006

Sorted sequence of numbers of the form A109043(A003173(k)), any k.

Cf. A003173.

h = {1, 2, 3, 7, 11, 19, 43, 67, 163} a = Union[Table[If[Mod[h[[n]] + 1, 2] == 0, h[[n]] + 1, 2*(h[[n]] + 1)], {n, 1, Length[h]}]]

keywords fini,full,less added by Assoc. Eds. of the OEIS, Jun 15 2010

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cp's OEIS Frontend

A168077 a(2n) = A129194(2n)/2; a(2n+1) = A129194(2n+1).

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A210530 T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2 n, k > 0, read by antidiagonals.

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A186421 Even numbers interleaved with repeated odd numbers.

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A066043 a(1) = 1; for m > 0, a(2m) = 2m, a(2m+1) = 4m+2.

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A227316 a(n) = n(n+1) if n == 0 or 1 (mod 4), otherwise a(n) = n(n+1)/2.

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A272104 Sum of the even numbers among the larger parts of the partitions of n into two parts.

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A210530 T(n,k) = (k + 3n - 2 - (k+n-2)(-1)^(k+n))/2 n, k > 0, read by antidiagonals.