A076100 -id:A076100 - cp's OEIS frontend

A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11, 7, 8, 9, 10, 11, 12, 13, 8, 9, 10, 11, 12, 13, 14, 15, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23

Offset: 1

Reinhard Zumkeller, May 24 2004

All numbers m occur ceiling(m/2) times, see A004526.

The LCM of the n-th row is A076100. - Michel Marcus, Mar 18 2018

			Triangle begins:
  1;
  2,  3;
  3,  4,  5;
  4,  5,  6,  7;
  5,  6,  7,  8,  9;
  6,  7,  8,  9, 10, 11;
  7,  8,  9, 10, 11, 12, 13;
  8,  9, 10, 11, 12, 13, 14, 15;
  9, 10, 11, 12, 13, 14, 15, 16, 17;
  ... - _Philippe Deléham_, Mar 30 2013

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Bruno Berselli, Illustration of the initial terms.
László Németh, On the Binomial Interpolated Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.7.8.
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.

Cf. A002024, A002260, A003056, A004526, A004736, A005408, A005843.
Cf. A016777, A016813, A016861, A037213, A076100, A078112, A093005.
Cf. A094728, A214604, A214661, A319556.
Cf. A128076 (rows reversed).

a094727 n k = n + k
a094727_row n = a094727_tabl !! (n-1)
a094727_tabl = iterate (\row@(h:_) -> (h + 1) : map (+ 2) row) [1]
-- Reinhard Zumkeller, Jul 22 2012

z:=12; &cat[ [m+n-1: m in [1..n] ]: n in [1..z] ];

Table[n + Range[0, n-1], {n, 12}]//Flatten (* Michael De Vlieger, Dec 16 2016 *)

from math import isqrt
def A094727(n): return ((a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(3-a)>>1)+n-1 # Chai Wah Wu, Jun 19 2025

flatten([[n+k for k in range(n)] for n in range(1,16)]) # G. C. Greubel, Mar 10 2024

T(n+1, k) = T(n, k) + 1 = T(n, k+1); T(n+1, k+1) = T(n, k) + 2.
T(n, n - A005843(k)) = A005843(n-k) for 0 <= k <= n/2.
T(n, n - A005408(k)) = A005408(n-k) for 0 <= k < n/2.
T(A005408(n), n) = A016777(n), n >= 0.
Sum_{k=1..n} T(n, k) = A000326(n) (row sums).
T(n, k) = A002024(n,k) + A002260(n,k) - 1. - Reinhard Zumkeller, Apr 27 2006
As a sequence rather than as a table: If m = floor((sqrt(8n-7)+1)/2), a(n) = n - m*(m-3)/2 - 1. - Carl R. White, Jul 30 2009
T(n, k) = n+k-1, n >= k >= 1. - Vincenzo Librandi, Nov 23 2009 [corrected by Klaus Brockhaus, Nov 23 2009]
T(n,k) = A037213((A214604(n,k) + A214661(n,k)) / 2). - Reinhard Zumkeller, Jul 25 2012
From Boris Putievskiy, Jan 16 2013: (Start)
a(n) = A002260(n) + A003056(n).
a(n) = i+t, where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)
From G. C. Greubel, Mar 10 2024: (Start)
T(3*n-3, n) = A016813(n-1).
T(4*n-4, n) = A016861(n-1).
Sum_{k=0..n-1} (-1)^k*T(n, k) = A319556(n).
Sum_{k=0..floor((n-1)/2)} T(n-k, k) = A093005(n).
Sum_{k=0..floor((n-1)/2)} (-1)^k*T(n-k, k) = A078112(n-1).
Sum_{j=1..n} (Sum_{k=0..n-1} T(j, k)) = A002411(n) (sum of n rows). (End)

A099996 a(n) = lcm{1, 2, ..., 2*n}.

Original entry on oeis.org

1, 2, 12, 60, 840, 2520, 27720, 360360, 720720, 12252240, 232792560, 232792560, 5354228880, 26771144400, 80313433200, 2329089562800, 144403552893600, 144403552893600, 144403552893600, 5342931457063200, 5342931457063200

Offset: 0

N. J. A. Sloane, Nov 20 2004

The prime number theorem implies that a(n) = e^(2n(1+o(1))) as n -> infinity. In other words, log(a(n))/n -> 2 as n -> infinity. (Sondow)

			The LCM of {1,2,3,4,5,6} is 60 and 6 = 2*3, so a(3) = 60.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003) 3335-3344.
Index entries for sequences related to lcm's

Bisection of A003418.
Cf. A076100, A093880.
Cf. A051173.

a099996 = foldl lcm 1 . enumFromTo 2 . (* 2)
-- Reinhard Zumkeller, Feb 11 2014

A099996 := proc(n)
        lcm(seq(i,i=1..2*n)) ;
end proc: # R. J. Mathar, Dec 14 2011

a(n) = lcm(vector(2*n, k, k)); \\ Michel Marcus, Mar 18 2018

More terms from Jonathan Sondow, Jan 17 2005

A069685 Denominators of coefficients in -log(1+x)log(1-x) power series.

Original entry on oeis.org

1, 12, 180, 1680, 12600, 166320, 2522520, 576576, 22054032, 465585120, 2560718160, 64250746560, 348024877200, 1124388064800, 4990906206000, 165032631878400, 350694342741600, 2599263952084800, 101515697684200800

Offset: 1

Benoit Cloitre, May 03 2002

Cf. A049281, A076100.

my(x='x+O('x^50), v=Vec(-log(1+x)*log(1-x))); apply(denominator, vector(#v\2, k, v[2*k-1])) \\ Michel Marcus, Feb 16 2021

a(n) = n*A076100(n) (conjectured). - F. Chapoton, Nov 09 2009

A211168 Exponent of alternating group An.

Original entry on oeis.org

1, 1, 3, 6, 30, 60, 420, 420, 1260, 2520, 27720, 27720, 360360, 360360, 360360, 360360, 6126120, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800

Offset: 1

Alexander Gruber, Jan 31 2013

a(n) is the smallest natural number m such that g^m = 1 for any g in An.

If m <= n, a m-cycle occurs in some permutation in An if and only if m is odd or m <= n - 2. The exponent is the LCM of the m's satisfying these conditions, leading to the formula below.

			For n = 7, lcm{1,...,5,7} = 420.

Alexander Gruber, Table of n, a(n) for n = 1..2308

Even entries given by the sequence A076100, or the odd entries in the sequence A003418.

The records of this sequence are a subsequence of A002809 and A126098.

for n in [1..40] do
Exponent(AlternatingGroup(n));
end for;

for n in [1..40] do
if n mod 2 eq 0 then
L := [1..n-1];
else
L := Append([1..n-2],n);
end if;
LCM(L);
end for;

Table[If[Mod[n, 2] == 0, LCM @@ Range[n - 1],
  LCM @@ Join[Range[n - 2], {n}]], {n, 1, 100}] (* or *)
a[1] = 1; a[2] = 1; a[3] = 3; a[n_] := a[n] =
  If[Mod[n, 2] == 0, LCM[a[n - 1], n - 2], LCM[a[n - 2], n - 3, n]]; Table[a[n], {n, 1, 40}]

a(n)=lcm(if(n%2,concat([2..n-2],n),[2..n-1])) \\ Charles R Greathouse IV, Mar 02 2014

Explicit:
a(n) = lcm{1, ..., n-1} if n is even.
     = lcm{1, ..., n-2, n} if n is odd.
Recursive:
Let a(1) = a(2) = 1 and a(3) = 3.  Then
a(n) = lcm{a(n-1), n-2} if n is even.
     = lcm{a(n-2), n-3, n} if n is odd.
a(n) = A003418(n)/(1 + [n in A228693]) for n > 1. - Charlie Neder, Apr 25 2019

A334224 Consider a graph as defined in A306302 formed from a row of n adjacent congruent squares with the diagonals of all possible rectangles; a(n) is the minimum edge length of the squares such that the vertices formed by all intersections have integer x and y coordinates.

Original entry on oeis.org

2, 6, 60, 420, 2520, 27720, 360360, 360360, 12252240, 232792560, 232792560, 5354228880, 26771144400, 80313433200, 2329089562800, 72201776446800, 144403552893600, 144403552893600, 5342931457063200

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 28 2020

nonn

			a(1) = 2 as for a single square, with its bottom left corner at the origin, with both diagonals drawn the intersection point of those lines is at (L/2,L/2) where L is the edge length. Thus L=2 for this to have integer coordinates.
a(2) = 6 as for two vertically adjacent squares the seven intersection points of the diagonals and shared internal edge have coordinates (L/3,4L/3),(L/2,3L/2),(2L/3,4L/3),(L/2,L),(L/3,2L/3),(L/2,L/2),(2L/3,2L/3). Thus L=6, the lowest common multiple of the denominators, for all these points to have integer coordinates.

Cf. A306302, A003418, A331755, A290132, A290131.

a(n) = A003418(2n-1) = A076100(n) for n>1.

cp's OEIS Frontend

A094727 Triangle read by rows: T(n,k) = n + k, 0 <= k < n, n >= 1.

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A099996 a(n) = lcm{1, 2, ..., 2*n}.

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A069685 Denominators of coefficients in -log(1+x)log(1-x) power series.

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PARI

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A211168 Exponent of alternating group An.

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Magma

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A334224 Consider a graph as defined in A306302 formed from a row of n adjacent congruent squares with the diagonals of all possible rectangles; a(n) is the minimum edge length of the squares such that the vertices formed by all intersections have integer x and y coordinates.

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