A216917 -id:A216917 - cp's OEIS frontend

A128501 a(n) = lcm{1 <= k <= n, gcd(k, 3) = 1}.

1, 1, 2, 2, 4, 20, 20, 140, 280, 280, 280, 3080, 3080, 40040, 40040, 40040, 80080, 1361360, 1361360, 25865840, 25865840, 25865840, 25865840, 594914320, 594914320, 2974571600, 2974571600, 2974571600, 2974571600, 86262576400

Offset: 0

Wolfdieter Lang, Apr 04 2007

Old name was: Denominators of partial sums for a series for Pi/(3*sqrt(3)).
The numerators are given in A128500. See the W. Lang link under A128500.
There appears to be a relationship between a(n) and b(n) = Denominator(3*HarmonicNumber(n)). For n=0..8, b(n)=a(n). For n=9..17, b(n)= 3*a(n). Starting at term 18, b(n)/a(n) = 1, 1, 1/5, 1/5, 1/5, 1/5, 1/5, 1, 1, 9, 9, 9, 9, 9, 9. - Gary Detlefs, Oct 12 2011 [adjusted to new definition by Peter Luschny, Oct 15 2012]

Cf. A003418, A216917, A217858.

A128501 := n -> ilcm(op(select(j->igcd(j,3) = 1,[$1..n]))):
seq(A128501(i),i=0..28); # Peter Luschny, Oct 15 2012

a[n_] := If[n == 0, 1, LCM @@ Select[Range[n], GCD[#, 3] == 1&]];
Array[a, 30, 0] (* Jean-François Alcover, Jun 14 2019, from Maple *)

def A128501(n): return lcm([j for j in (1..n) if gcd(j,3) == 1])
[A128501(n) for n in (0..28)]  # Peter Luschny, Oct 15 2012

a(n+1) = denominator(r(n)) with the rationals r(n):=Sum_{k=0..n} ((-1)^k)*S(k,1)/(k+1) with Chebyshev's S-Polynomials S(n,1)=[1,1,0,-1,-1,0] periodic sequence with period 6. See A010892.

New name and 1 prepended by Peter Luschny, Oct 15 2012

A217858 Odd part of lcm(1,2,3,...,n).

Original entry on oeis.org

1, 1, 3, 3, 15, 15, 105, 105, 315, 315, 3465, 3465, 45045, 45045, 45045, 45045, 765765, 765765, 14549535, 14549535, 14549535, 14549535, 334639305, 334639305, 1673196525, 1673196525, 5019589575, 5019589575, 145568097675, 145568097675, 4512611027925, 4512611027925, 4512611027925

Offset: 1

Joerg Arndt, Oct 13 2012

nonn

			a(8) is the least common multiple of {1,3,5,7}.

Cf. A003418, A025547, A128501, A216917.

A217858 := n -> ilcm(op(select(j->j mod 2 = 1, [$1..n]))):
seq(A217858(i), i = 1..33); # Peter Luschny, Oct 15 2012

a[n_] := LCM @@ Select[Range[n], OddQ]; Array[a, 33] (* Jean-François Alcover, Jun 27 2019 *)

A003418(n) = lcm(vector(n,k, k));
A000265(n) = n/2^valuation(n,2);
a(n) = A000265( A003418(n) );

def A217858(n): return lcm([j for j in (1..n) if is_odd(j)])
[A217858(n) for n in (1..33)]  # Peter Luschny, Oct 15 2012

a(n) = A000265( A003418(n) ).
a(n) = lcm_{1 <= k <= n, k odd}. - Peter Luschny, Oct 15 2012
a(n) = A025547(floor((n+1)/2)). - Georg Fischer, Nov 29 2022

A216915 T(n, k) = Product{1<=j<=n, gcd(j,k)=1 | j} / lcm{1<=j<=n, gcd(j,k)=1 | j} for n >= 0, k >= 1, square array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 2, 1, 1, 1, 1, 12, 1, 2, 1, 1, 1, 1, 1, 48, 1, 2, 1, 2, 1, 1, 1, 1, 144, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1440, 3, 8, 1, 12, 1, 2, 1, 1, 1, 1, 1440, 3, 8, 1, 12, 1, 2, 1, 1, 1, 1, 1, 17280, 3, 80

Offset: 1

Peter Luschny, Oct 02 2012

T(n,k) = Product(R(n,k))/lcm(R(n,k)) where R(n,k) is the set of all integers up to n that are relatively prime to k.

T(n,k) = A216919(n,k)/A216917(n,k).

			   k |n=0  1  2  3  4  5  6  7  8   9   10
  ---+------------------------------------
   1 |  1  1  1  1  2  2 12 12 48 144 1440
   2 |  1  1  1  1  1  1  1  1  1   3    3
   3 |  1  1  1  1  2  2  2  2  8   8   80
   4 |  1  1  1  1  1  1  1  1  1   3    3
   5 |  1  1  1  1  2  2 12 12 48 144  144
   6 |  1  1  1  1  1  1  1  1  1   1    1
   7 |  1  1  1  1  2  2 12 12 48 144 1440
   8 |  1  1  1  1  1  1  1  1  1   3    3
   9 |  1  1  1  1  2  2  2  2  8   8   80
  10 |  1  1  1  1  1  1  1  1  1   3    3
  11 |  1  1  1  1  2  2 12 12 48 144 1440
  12 |  1  1  1  1  1  1  1  1  1   1    1
  13 |  1  1  1  1  2  2 12 12 48 144 1440

def A216915(n, k):
    def R(n, k): return [j for j in (1..n) if gcd(j, k) == 1]
    return mul(R(n,k))/lcm(R(n, k))
for k in (1..13): [A216915(n, k) for n in (0..10)]

For n > 0:
A(n,1) = A025527(n);
A(4,n) = A000034(n);
A(n,n) = A128247(n).

cp's OEIS Frontend

A128501 a(n) = lcm{1 <= k <= n, gcd(k, 3) = 1}.

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A217858 Odd part of lcm(1,2,3,...,n).

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A216915 T(n, k) = Product{1<=j<=n, gcd(j,k)=1 | j} / lcm{1<=j<=n, gcd(j,k)=1 | j} for n >= 0, k >= 1, square array read by antidiagonals.

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