A002944 -id:A002944 - cp's OEIS frontend

A387027 a(n) = lcm({1, 2, ..., n}) * (n + 1) / n for n > 0, a(0) = 1.

1, 2, 3, 8, 15, 72, 70, 480, 945, 2800, 2772, 30240, 30030, 388080, 386100, 384384, 765765, 12972960, 12932920, 245044800, 244432188, 243877920, 243374040, 5587021440, 5577321750, 27841990176, 27800803800, 83288004800, 83181770100, 2409402996000, 2406725881560

Offset: 0

Peter Luschny, Aug 17 2025

nonn

Paolo Xausa, Table of n, a(n) for n = 0..2000

Cf. A003418.

A387027 := n -> local k; ifelse(n = 0, 1, ((n+1) * lcm(seq(k, k = 1..n))) / n):
seq(A387027(n), n = 0..30);

A387027[n_] := If[n == 0, 1, Quotient[(n + 1) LCM @@ Range[1, n], n]];
Table[A387027[n], {n, 0, 30}]

from math import lcm
def A387027(n): return (n+1)*lcm(*range(1,n+1))//n if n else 1 # Chai Wah Wu, Aug 17 2025

a(n) = A003418(n) * (n + 1) / n for n >= 1.

a(n) = (n+1)*A002944(n). - R. J. Mathar, Aug 19 2025

A118975 Triangular array: row n consists of M times n-th-order Farey fractions, where M is the LCM of their denominators.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 3, 4, 6, 8, 9, 12, 12, 15, 20, 24, 30, 36, 40, 45, 48, 60, 10, 12, 15, 20, 24, 30, 36, 40, 45, 48, 50, 60, 60, 70, 84, 105, 120, 140, 168, 180, 210, 240, 252, 280, 300, 315, 336, 350, 360, 420, 105, 120, 140, 168, 210, 240

Offset: 1

Clark Kimberling, May 07 2006

The multiplier M(n) = A003418(n). Column 1 is A002944(n) = (1/n)*lcm(1,2,...,n). The numbers in row n are relatively prime (unlike the related array A093594).

			Row 4 is 12*(1/4, 1/3, 1/2, 2/3, 3/4, 1).
First 5 rows:
   1
   1  2
   2  3  4  6
   3  4  6  8  9 12
  12 15 20 24 30 36 40 45 48 60.

Cf. A003418.

A119936 Least common multiple (LCM) of denominators of the rows of the triangle of rationals A119935/A119932.

Original entry on oeis.org

1, 8, 108, 576, 18000, 21600, 1234800, 5644800, 57153600, 63504000, 8452382400, 9220780800, 1688171284800, 1818030614400, 1947889944000, 8310997094400, 2551995545299200, 2702112930316800, 1029655143835718400

Offset: 1

Wolfdieter Lang, Jul 20 2006

The triangle of rationals is the matrix cube of the matrix with elements a(i,j) = 1/i if j <= i, 0 if j > i.

Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012.

Distinct from A246498.

A027447(i,j)= a(i)* A119935(i,j)/A119932(i,j) .
a(n) = lcm_{m=1..n} seq(A119932(n,m)), n >= 1.
a(n)/n^3 = A027451(n) = A002944(n)^2 (the second equation is a conjecture).
a(n)/n^3 = (A099946(n)*(n-1))^2, n >= 2 (from the conjecture).

A174554 Smallest k > 2 such that 2|k, 3|k+1, 4|k+2,..., n|k+n-2.

Original entry on oeis.org

4, 8, 14, 62, 62, 422, 842, 2522, 2522, 27722, 27722, 360362, 360362, 360362, 720722, 12252242, 12252242, 232792562, 232792562, 232792562, 232792562, 5354228882, 5354228882, 26771144402, 26771144402, 80313433202, 80313433202

Offset: 2

Michel Lagneau, Mar 22 2010

nonn

We solve the system of n+1 equations : k==2 (mod 2), k==2 (mod 3),...,k==2 (mod n), and then the solutions are k== 2 mod (lcm(2,3,4,...,n)) where lcm(k) is the least common multiple of{1, 2, ..., k}(A003418) .

			a(2) = 4 because 2|4;
a(3) = 8 because 2|8 and 3|9;
a(4) = 14 because 2|14, 3|15 and 4|16;
a(5) = 62 because 2|62, 3|63, 4|64 and 5|65;
a(6) = 62 because 2|62, 3|63, 4|64, 5|65 and 6|66.

Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, arXiv:0906.2295 [math.NT], 2009.
Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, Amer. Math. Monthly, 116 (2009), 836-839.
Eric Weisstein's World of Mathematics, Least Common Multiple

Cf. A002944, A003990, A051173, A000793, A003418, A048691.

with(numtheory):q:=2:for k from 2 to 100 do :q1:= lcm(q,k):q2 :=2+q1 :print(q2): q :=q1 :od :

a(n) = 2 + lcm(2,3,4,...,n) = A003418(n) + 2.

A265286 Minimal number of pieces of a cake such that they can be distributed equally among k guests for any k=1,2,...,n.

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 14, 16, 19

Offset: 1

Max Alekseyev, Dec 06 2015

An equivalent formulation in terms of integers: after multiplying by the LCM of the denominators, a(n) is the minimal cardinality M of a multiset S of positive integers which can be partitioned into k multisets with equal sums for all k = 1, ..., n.
A subsidiary problem: Look at all the multisets S for a given value of n that have M = a(n) elements, and let g(n) denote the minimal value of the largest element of any such S. The initial values of g(n) for n=1..6 are 1, 1, 2, 3, 12, 10, which suggests that g(n) might equal A002944(n) = lcm{1..n}/n. Multisets associated with these values of g(n) are {1}, {1,1}, {1,1,2,2}, {1,1,1,3,3,3}, {1,2,3,5,7,8,10,12,12}, {2,2,3,4,4,5,5,7,8,10,10}. - Max Alekseyev and N. J. A. Sloane, Jan 25 2016
a(n) <= a(n-1) + n - A032742(n).
Bounds for later terms: a(10)<=22, a(11)<=28, a(12)<=30, a(13)<=42, a(14)<=49 (see dxdy.ru link).
For n>=5, a(n) >= 2n-1. This bound holds even if we restrict k to {n-2,n-1,n} only.
We could also ask about the smallest piece in any of the multisets S. For n=6, the minimum smallest piece in an 11-piece solution is 1/120, as in [1/120, 1/40, 1/30, 7/120, 3/40, 11/120, 13/120, 1/8, 17/120, 1/6, 1/6]. But this is a different question from finding g(n). - Max Alekseyev, Jan 24 2016

			For n=5, the minimal number of pieces is 9. Taking the cake size to be 1, a set of possible pieces is {1/60, 1/30, 1/20, 1/12, 7/60, 2/15, 1/6, 1/5, 1/5}, so that for 1 <= k <= 5 guests we have the following partitions:
k=1: 1/60 + 1/30 + 1/20 + 1/12 + 7/60 + 2/15 + 1/6 + 1/5 + 1/5 [ = 1 ]
k=2: 1/60 + 1/30 + 1/20 + 1/12 + 7/60 + 1/5 = 2/15 + 1/6 + 1/5 [ = 1/2 ]
k=3: 1/60 + 7/60 + 1/5 = 1/30 + 1/20 + 1/12 + 1/6 = 2/15 + 1/5 [ = 1/3 ]
k=4: 1/60 + 1/30 + 1/5 = 1/20 + 1/5 = 1/12 + 1/6 = 7/60 + 2/15 [ = 1/4 ]
k=5: 1/60 + 1/20 + 2/15 = 1/30 + 1/6 = 1/12 + 7/60 = 1/5 = 1/5 [ = 1/5 ]
Another solution for n=5 is {1/120, 1/24, 7/120, 11/120, 13/120, 17/120, 19/120, 23/120, 1/5}. Notice that denominators here are not bounded by A003418(5)=60.
Examples corresponding to the formulation in terms of multisets described in the comments:
n=1: {1},
n=2: (1,1)/2,
n=3: {1,1,2,2}/6,
n=4: {1,1,1,3,3,3}/12,
n=5: {1,2,3,5,7,8,10,12,12}/60 (as above),
n=6: {2,2,3,4,4,5,5,7,8,10,10}/60
n=7: {1,11,15,19,21,25,29,31,35,39,41,45,49,59}/420,
n=8: {17,23,25,32,37,38,47,52,53,58,67,68,73,80,82,88}/840,
n=9: {21,56,69,85,95,101,115,119,120,130,150,155,160,161,165,179,185,195,259}/2520.

Glinka et al., Minimal possible cardinality of a (a1,...,ak)-distributable multiset, Mathoverflow, 2015.
Multiple authors, Discussion at dxdy.ru (in Russian)

Cf. A002944, A003418.

Values a(1)-a(9) are established at dxdy.ru (see link)

Edited by N. J. A. Sloane, Jan 25 2016

cp's OEIS Frontend

A387027 a(n) = lcm({1, 2, ..., n}) * (n + 1) / n for n > 0, a(0) = 1.

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A118975 Triangular array: row n consists of M times n-th-order Farey fractions, where M is the LCM of their denominators.

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A119936 Least common multiple (LCM) of denominators of the rows of the triangle of rationals A119935/A119932.

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A174554 Smallest k > 2 such that 2|k, 3|k+1, 4|k+2,..., n|k+n-2.

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A265286 Minimal number of pieces of a cake such that they can be distributed equally among k guests for any k=1,2,...,n.

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