A128501 -id:A128501 - cp's OEIS frontend

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

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).

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

A128500 Numerators of partial sums for a series for Pi/(3*sqrt(3)).

Original entry on oeis.org

1, 1, 1, 3, 11, 11, 97, 159, 159, 187, 1777, 1777, 26181, 23321, 23321, 51647, 797919, 797919, 16521821, 15228529, 15228529, 16404249, 351431887, 351431887, 1876142299, 1761735699, 1761735699, 1867970399, 51196569971, 51196569971

Offset: 0

Wolfdieter Lang Apr 04 2007

The denominators are given in A128501.

The limit n -> infinity of the rationals r(n) defined below is Pi/(3*sqrt(3)).

			Rationals: [1, 1/2, 1/2, 3/4, 11/20, 11/20, 97/140, 159/280, 159/280, 187/280,...]
Pi/(3*sqrt(3))=+1/1 -1/2 +1/4 -1/5 +1/7 -1/8 +1/10 -1/11 +1/13 -+

W. Lang, Rationals and limit.

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

cp's OEIS Frontend

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

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

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A128500 Numerators of partial sums for a series for Pi/(3*sqrt(3)).

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