A109051 -id:A109051 - cp's OEIS frontend

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

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).

A106611 a(n) = numerator of n/(n+10).

Original entry on oeis.org

0, 1, 1, 3, 2, 1, 3, 7, 4, 9, 1, 11, 6, 13, 7, 3, 8, 17, 9, 19, 2, 21, 11, 23, 12, 5, 13, 27, 14, 29, 3, 31, 16, 33, 17, 7, 18, 37, 19, 39, 4, 41, 21, 43, 22, 9, 23, 47, 24, 49, 5, 51, 26, 53, 27, 11, 28, 57, 29, 59, 6, 61, 31, 63, 32, 13, 33, 67, 34, 69, 7, 71, 36, 73, 37, 15, 38, 77, 39

Offset: 0

N. J. A. Sloane, May 15 2005

A strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n,m >= 1. It follows that a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 17 2019

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Wikipedia, Quasi-polynomial.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-1).

Cf. A023900, A109051, A303367.

Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106612 (k = 7 thru 11), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

List([0..80],n->NumeratorRat(n/(n+10))); # Muniru A Asiru, Feb 18 2019

[Numerator(n/(n+10)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

f[n_]:=Numerator[n/(n+10)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)

[lcm(n,10)/10 for n in range(0, 79)] # Zerinvary Lajos, Jun 07 2009

From R. J. Mathar, Apr 18 2011: (Start)
a(n) = A109051(n)/10.
Dirichlet g.f.: zeta(s-1)*(1 - 4/5^s - 1/2^s + 4/10^s).
Multiplicative with a(2^e) = 2^max(0,e-1), a(5^e) = 5^max(0,e-1), a(p^e) = p^e if p = 3 or p >= 7. (End)
From Peter Bala, Feb 17 2019: (Start)
a(n) = numerator(n/((n + 2)*(n + 5))).
a(n) = n/b(n), where b(n) = [1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, ...] is a purely periodic sequence of period 10. Thus a(n) is a quasi-polynomial in n.
If gcd(n,m) = 1 then a( a(n)*a(m) ) = a(a(n)) * a(a(m)), a( a(a(n))*a(a(m)) ) = a(a(a(n))) * a(a(a(m))) and so on.
O.g.f.: Sum_{d divides 10} A023900(d)*x^d/(1 - x^d)^2 = x/(1 - x)^2 - x^2/(1 - x^2)^2 - 4*x^5/(1 - x^5)^2 + 4*x^10/(1 - x^10)^2.
(End)
Sum_{k=1..n} a(k) ~ (63/200) * n^2. - Amiram Eldar, Nov 25 2022

A070291 a(n) = lcm(10,n)/gcd(10,n).

Original entry on oeis.org

10, 5, 30, 10, 2, 15, 70, 20, 90, 1, 110, 30, 130, 35, 6, 40, 170, 45, 190, 2, 210, 55, 230, 60, 10, 65, 270, 70, 290, 3, 310, 80, 330, 85, 14, 90, 370, 95, 390, 4, 410, 105, 430, 110, 18, 115, 470, 120, 490, 5, 510, 130, 530, 135, 22, 140, 570, 145, 590, 6

Offset: 1

Amarnath Murthy, May 10 2002

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

Cf. A070290, A070292, A070293, A109013, A109051.

Table[LCM[10,n]/GCD[10,n],{n,80}] (* Harvey P. Dale, Nov 01 2013 *)

a(n) = A109051(n) / A109013(n). - R. J. Mathar, Feb 12 2019
a(n) = 2*a(n-10) - a(n-20). - R. J. Mathar, Feb 12 2019
Sum_{k=1..n} a(k) ~ (101/40)*n^2. - Amiram Eldar, Oct 07 2023

cp's OEIS Frontend

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

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A106611 a(n) = numerator of n/(n+10).

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A070291 a(n) = lcm(10,n)/gcd(10,n).

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