A217858 -id:A217858 - 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

A216917 Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 12, 3, 2, 1, 1, 60, 3, 2, 1, 1, 1, 60, 15, 4, 3, 2, 1, 1, 420, 15, 20, 3, 6, 1, 1, 1, 840, 105, 20, 15, 12, 1, 2, 1, 1, 2520, 105, 140, 15, 12, 1, 6, 1, 1, 1, 2520, 315, 280, 105, 12, 5, 12, 3, 2, 1, 1, 27720, 315, 280, 105, 84

Offset: 1

Peter Luschny, Oct 02 2012

T(N,n) is the least common multiple of all integers up to N that are relatively prime to n.

Replacing LCM in the definition with "product" gives the Gauss factorial A216919.

			   n | N=0 1 2 3  4  5  6   7   8    9   10
-----+-------------------------------------
   1 |   1 1 2 6 12 60 60 420 840 2520 2520
   2 |   1 1 1 3  3 15 15 105 105  315  315
   3 |   1 1 2 2  4 20 20 140 280  280  280
   4 |   1 1 1 3  3 15 15 105 105  315  315
   5 |   1 1 2 6 12 12 12  84 168  504  504
   6 |   1 1 1 1  1  5  5  35  35   35   35
   7 |   1 1 2 6 12 60 60  60 120  360  360
   8 |   1 1 1 3  3 15 15 105 105  315  315
   9 |   1 1 2 2  4 20 20 140 280  280  280
  10 |   1 1 1 3  3  3  3  21  21   63   63
  11 |   1 1 2 6 12 60 60 420 840 2520 2520
  12 |   1 1 1 1  1  5  5  35  35   35   35
  13 |   1 1 2 6 12 60 60 420 840 2520 2520

t[, 0] = 1; t[n, k_] := LCM @@ Select[Range[k], CoprimeQ[#, n]&]; Table[t[n - k + 1, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

def A216917(N, n):
    return lcm([j for j in (1..N) if gcd(j, n) == 1])
for n in (1..13): [A216917(N,n) for N in (0..10)]

For n > 0:
A(n,1) = A003418(n);
A(n,2^k) = A217858(n) for k > 0;
A(n,3^k) = A128501(n-1) for k > 0;
A(2,n) = A000034(n);
A(3,n) = A129203(n-1);
A(4,n) = A129197(n-1);
A(n,n) = A038610(n);
A(floor(n/2),n) = A124443(n);
A(n,1)/A(n,n) = A064446(n);
A(n,1)/A(n,2) = A053644(n).

cp's OEIS Frontend

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

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