A109047 -id:A109047 - cp's OEIS frontend

A060789 a(n) = n / (gcd(n,2) * gcd(n,3)).

1, 1, 1, 2, 5, 1, 7, 4, 3, 5, 11, 2, 13, 7, 5, 8, 17, 3, 19, 10, 7, 11, 23, 4, 25, 13, 9, 14, 29, 5, 31, 16, 11, 17, 35, 6, 37, 19, 13, 20, 41, 7, 43, 22, 15, 23, 47, 8, 49, 25, 17, 26, 53, 9, 55, 28, 19, 29, 59, 10, 61, 31, 21, 32, 65, 11, 67, 34, 23, 35, 71, 12, 73, 37, 25, 38, 77

Offset: 1

Len Smiley, Apr 26 2001

a(n+2) is absolute value of numerator of determinant of n X n matrix with M(i,j) = 2/(i(i+1)) if i=j otherwise 1. - Alexander Adamchuk, May 19 2006
Numerator of n/(n+6). - Gerry Martens, Aug 06 2015
In addition to being multiplicative, this sequence is also a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n, m)) for n, m >= 1. In particular, it follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 20 2019
Connected to the Diophantine equation x^2 + (x + 1)^2 + … + (x + k)^2 = C, C an integer, x >= 0, k >= 0. Rewritten it is (k + 1)*x^2 + k*(k+1)*x + k*(k + 1)*(2*k + 1)/6 = C. The existence of its solutions depends on gcd(k + 1, k*(k + 1), k*(k + 1)*(2*k + 1)/6). - Ctibor O. Zizka, Oct 04 2023

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

Cf. A011782, A109047, A146535, A220466.

Cf. other sequences given by the formula n/gcd(n,k) = numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([1..80],n->n/(Gcd(n,2)*Gcd(n,3))); # Muniru A Asiru, Feb 20 2019

[n/(Gcd(n,2)*Gcd(n,3)) : n in [1..100]]; // Wesley Ivan Hurt, Aug 06 2015

I:=[1,1,1,2,5,1,7,4,3,5,11,2]; [n le 12 select I[n] else 2*Self(n-6)-Self(n-12): n in [1..80]]; // Vincenzo Librandi, Aug 07 2015

a := proc(n): if n = 1 then 1 else denom((4*n-6)/n) fi: end: seq(a(n), n=1..77); # Johannes W. Meijer, Dec 19 2012

Numerator[Table[(-1)^(n+1) Det[ DiagonalMatrix[ Table[ 2/(i(i+1)) - 1, {i, 1, n-2} ] ] + 1 ], {n, 30} ]] (* Alexander Adamchuk, May 19 2006 *)
Table[Numerator[(n+3)/(n+2)/(n+1)/n],{n,60}] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)
Table[n/(GCD[n, 2] GCD[n, 3]), {n, 100}] (* Wesley Ivan Hurt, Aug 06 2015 *)
LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1}, {1,1,1,2,5,1,7,4,3,5,11,2}, 80] (* Vincenzo Librandi, Aug 07 2015 *)

a(n) = { n / (gcd(n, 2) * gcd(n, 3)) } \\ Harry J. Smith, Jul 11 2009

[lcm(n,6)/6 for n in range(1, 78)] # Zerinvary Lajos, Jun 07 2009

G.f.: x*(1 + x + x^2 + 2*x^3 + 5*x^4 + x^5 + 5*x^6 + 2*x^7 + x^8 + x^9 + x^10)/(1 - x^6)^2.
Multiplicative with a(2^e)=2^(e-1), a(3^e)=3^(e-1), a(p^e)=p^e, p>3. - Vladeta Jovovic, Sep 09 2004
a(n) = Numerator[(-1)^(n+1)*Det[DiagonalMatrix[Table[2/(i(i+1))-1, {i,1,n-2}]]+1]], n>2. - Alexander Adamchuk, May 19 2006
a(n) divides n. a(6k) = k for integer k>0. a(p^k) = p^k for prime p>3 and integer k>0. - Alexander Adamchuk, Sep 20 2006
From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109047(n)/6.
Dirichlet g.f. zeta(s-1)*(1-1/2^s-2/3^s+2/6^s). (End)
a(n) = denominator((4*n-6)/n), n >= 2, with a(1) = 1. - Johannes W. Meijer, Dec 19 2012
a((2*n-1)*2^p) = A011782(p)*A146535(n), p >= 0. - Johannes W. Meijer, Feb 06 2013
a(n) = 2*a(n-6) - a(n-12) for n >= 12. - Robert Israel, Aug 06 2015
a(n) = gcd((n-1)*n*(n+1)/6, n). - Lechoslaw Ratajczak, Feb 16 2017
From Peter Bala, Feb 13 2019: (Start)
a(n) = n/gcd(n,n + 6) = n/gcd(n,6).
a(n) = n/b(n), where b(n) is the purely periodic sequence [1,2,3,2,1,6,...] with period 6.
a(n) is a quasi-polynomial in n: a(6*n+1) = 6*n + 1; a(6*n+2) = 3*n + 1; a(6*n+3) = 2*n + 1; a(6*n+4) = 3*n + 2; a(6*n+5) = 6*n + 5; a(6*n) = n.
a(n) = numerator(n/(n + 6)); a(n) = denominator((n + 6)/n).
(End)
Sum_{k=1..n} a(k) ~ 7*n^2/24. - Vaclav Kotesovec, Aug 09 2022
From Ctibor O. Zizka, Oct 04 2023: (Start)
For k >=0, a(k) = gcd(k + 1, k*(k + 1), k*(k + 1)*(2*k + 1)/6).
If (k mod 6) = 0 or 4 then a(k) = (k + 1).
If (k mod 6) = 1 or 3 then a(k) = (k + 1)/2.
If (k mod 6) = 2 then a(k) = (k + 1)/3.
If (k mod 6) = 5 then a(k) = (k + 1)/6. (End)

A109042 Square array read by antidiagonals: A(n, k) = lcm(n, k) for n >= 0, k >= 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 4, 3, 4, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 6, 15, 4, 15, 6, 7, 0, 0, 8, 14, 6, 20, 20, 6, 14, 8, 0, 0, 9, 8, 21, 12, 5, 12, 21, 8, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 10, 9, 8, 35, 6, 35, 8, 9, 10, 11, 0

Offset: 0

Mitch Harris, Jun 18 2005

			Seen as an array:
  [0] 0, 0,  0,  0,  0,  0,  0,  0,  0,  0, ...
  [1] 0, 1,  2,  3,  4,  5,  6,  7,  8,  9, ...
  [2] 0, 2,  2,  6,  4, 10,  6, 14,  8, 18, ...
  [3] 0, 3,  6,  3, 12, 15,  6, 21, 24,  9, ...
  [4] 0, 4,  4, 12,  4, 20, 12, 28,  8, 36, ...
  [5] 0, 5, 10, 15, 20,  5, 30, 35, 40, 45, ...
  [6] 0, 6,  6,  6, 12, 30,  6, 42, 24, 18, ...
  [7] 0, 7, 14, 21, 28, 35, 42,  7, 56, 63, ...
  [8] 0, 8,  8, 24,  8, 40, 24, 56,  8, 72, ...
  [9] 0, 9, 18,  9, 36, 45, 18, 63, 72,  9, ...
.
Seen as a triangle T(n, k) = lcm(n - k, k).
  [0] 0;
  [1] 0, 0;
  [2] 0, 1,  0;
  [3] 0, 2,  2,  0;
  [4] 0, 3,  2,  3,  0;
  [5] 0, 4,  6,  6,  4,  0;
  [6] 0, 5,  4,  3,  4,  5, 0;
  [7] 0, 6, 10, 12, 12, 10, 6,  0;
  [8] 0, 7,  6, 15,  4, 15, 6,  7, 0;
  [9] 0, 8, 14,  6, 20, 20, 6, 14, 8, 0;

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Rows A000027, A109043, A109044, A109045, A109046, A109047, A109048, A109049, A109050, A109051, A109052, A109053, A006580 (row sums of triangle), A001477 (main diagonal, central terms).

Variants: A003990 is (1, 1) based, A051173 (T(n,k) = lcm(n,k)).

T := (n, k) -> ilcm(n - k, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Mar 24 2025

lcm(n, k) = n*k / gcd(n, k) for (n, k) != (0, 0).

A283443 a(n) = lcm(n,6) / gcd(n,6).

Original entry on oeis.org

0, 6, 3, 2, 6, 30, 1, 42, 12, 6, 15, 66, 2, 78, 21, 10, 24, 102, 3, 114, 30, 14, 33, 138, 4, 150, 39, 18, 42, 174, 5, 186, 48, 22, 51, 210, 6, 222, 57, 26, 60, 246, 7, 258, 66, 30, 69, 282, 8, 294, 75, 34, 78, 318, 9, 330, 84, 38, 87, 354, 10, 366, 93, 42

Offset: 0

Colin Barker, Mar 07 2017

If n == 2 or 4 (mod 6) then a(n) = 3*n/2; if n == 3 (mod 6) then a(n) = 2*n/3; if n == 1 or 5 (mod 6) then a(n) = 6*n; otherwise, a(n) = n/6. Examples: n = 50 = 6*8+2, a(50) = 3*50/2 = 75; n = 23 = 6*3+5, a(23) = 6*23 = 138. - Bruno Berselli, Mar 08 2017

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

Cf. A064680, A130724, A188134, A283442, A283444.

Table[LCM[n, 6] / GCD[n, 6], {n,0,63}] (* Indranil Ghosh, Mar 08 2017 *)

concat(0, Vec(x*(6 + 3*x + 2*x^2 + 6*x^3 + 30*x^4 + x^5 + 30*x^6 + 6*x^7 + 2*x^8 + 3*x^9 + 6*x^10) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2) + O(x^100)))

{for (n=0, 63, print1((lcm(n, 6) / gcd(n, 6)),", "))}; \\ Indranil Ghosh, Mar 08 2017

G.f.: x*(6 + 3*x + 2*x^2 + 6*x^3 + 30*x^4 + x^5 + 30*x^6 + 6*x^7 + 2*x^8 + 3*x^9 + 6*x^10) / ((1 - x)^2*(1 + x)^2*(1 - x + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-6) - a(n-12) for n>11.
a(n) = A109047(n)/A089128(n). - R. J. Mathar, Feb 12 2019
Sum_{k=1..n} a(k) ~ (95/72)*n^2. - Amiram Eldar, Oct 07 2023

A353091 Irregular triangle read by rows: T(n, k) is the number of n-step closed walks on the hexagonal lattice having algebraic area k.

Original entry on oeis.org

1, 6, 0, 6, 66, 0, 12, 0, 150, 0, 30, 1020, 0, 420, 0, 84, 0, 6, 0, 3444, 0, 1302, 0, 252, 0, 42, 19890, 0, 11952, 0, 4284, 0, 984, 0, 216, 0, 24, 0, 82062, 0, 42972, 0, 14814, 0, 4248, 0, 990, 0, 216, 0, 18, 449976, 0, 327420, 0, 158970, 0, 57180, 0, 18780, 0, 5190, 0, 1350, 0, 270, 0, 30

Offset: 0

Andrey Zabolotskiy, Apr 22 2022

Rows 0 and 2 have 1 element each; row 1 is empty; for n > 2, we have 0 <= k <= A069813(n).

Rows can be extended to negative k with T(n, -k) = T(n, k). Sums of such extended rows give A002898.

			The triangle begins:
[1]
[]
[6]
[0, 6]
[66, 0, 12]
[0, 150, 0, 30]
[1020, 0, 420, 0, 84, 0, 6]
[0, 3444, 0, 1302, 0, 252, 0, 42]
[19890, 0, 11952, 0, 4284, 0, 984, 0, 216, 0, 24]
[0, 82062, 0, 42972, 0, 14814, 0, 4248, 0, 990, 0, 216, 0, 18]
[449976, 0, 327420, 0, 158970, 0, 57180, 0, 18780, 0, 5190, 0, 1350, 0, 270, 0, 30]
...

Cf. A069813 (greatest area), A002898 (all closed walks), A352838 (square lattice).

For n > 1, row n seems to end with A109047(n).

cp's OEIS Frontend

A060789 a(n) = n / (gcd(n,2) * gcd(n,3)).

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A109042 Square array read by antidiagonals: A(n, k) = lcm(n, k) for n >= 0, k >= 0.

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A283443 a(n) = lcm(n,6) / gcd(n,6).

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A353091 Irregular triangle read by rows: T(n, k) is the number of n-step closed walks on the hexagonal lattice having algebraic area k.

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