A099996 -id:A099996 - cp's OEIS frontend

A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.

1, 1, 2, 6, 12, 60, 60, 420, 840, 2520, 2520, 27720, 27720, 360360, 360360, 360360, 720720, 12252240, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800

Offset: 0

Roland Anderson (roland.anderson(AT)swipnet.se)

The minimal exponent of the symmetric group S_n, i.e., the least positive integer for which x^a(n)=1 for all x in S_n. - Franz Vrabec, Dec 28 2008
Product over all primes of highest power of prime less than or equal to n. a(0) = 1 by convention.
Also smallest number whose set of divisors contains an n-term arithmetic progression. - Reinhard Zumkeller, Dec 09 2002
An assertion equivalent to the Riemann hypothesis is: | log(a(n)) - n | < sqrt(n) * log(n)^2. - Lekraj Beedassy, Aug 27 2006. (This is wrong for n = 1 and n = 2. Should "for n large enough" be added? - Georgi Guninski, Oct 22 2011)
Corollary 3 of Farhi gives a proof that a(n) >= 2^(n-1). - Jonathan Vos Post, Jun 15 2009
Appears to be row products of the triangle T(n,k) = b(A010766) where b = A130087/A130086. - Mats Granvik, Jul 08 2009
Greg Martin (see link) proved that "the product of the Gamma function sampled over the set of all rational numbers in the open interval (0,1) whose denominator in lowest terms is at most n" equals (2*Pi)^(1/2)*a(n)^(-1/2). - Jonathan Vos Post, Jul 28 2009
a(n) = lcm(A188666(n), A188666(n)+1, ..., n). - Reinhard Zumkeller, Apr 25 2011
a(n+1) is the smallest integer such that all polynomials a(n+1)*(1^i + 2^i + ... + m^i) in m, for i=0,1,...,n, are polynomials with integer coefficients. - Vladimir Shevelev, Dec 23 2011
It appears that A020500(n) = a(n)/a(n-1). - Asher Auel, corrected by Bill McEachen, Apr 05 2024
n-th distinct value = A051451(n). - Matthew Vandermast, Nov 27 2009
a(n+1) = least common multiple of n-th row in A213999. - Reinhard Zumkeller, Jul 03 2012
For n > 2, (n-1) = Sum_{k=2..n} exp(a(n)*2*i*Pi/k). - Eric Desbiaux, Sep 13 2012
First column minus second column of A027446. - Eric Desbiaux, Mar 29 2013
For n > 0, a(n) is the smallest number k such that n is the n-th divisor of k. - Michel Lagneau, Apr 24 2014
Slowest growing integer > 0 in Z converging to 0 in Z^ when considered as profinite integer. - Herbert Eberle, May 01 2016
What is the largest number of consecutive terms that are all equal? I found 112 equal terms from a(370261) to a(370372). - Dmitry Kamenetsky, May 05 2019
Answer: there exist arbitrarily long sequences of consecutive terms with the same value; also, the maximal run of consecutive terms with different values is 5 from a(1) to a(5) (see link Roger B. Eggleton). - Bernard Schott, Aug 07 2019
Related to the inequality (54) in Ramanujan's paper about highly composite numbers A002182, also used in A199337: a(A329570(m))^2 is a (not minimal) bound above which all highly composite numbers are divisible by m, according to the right part of that inequality. - M. F. Hasler, Jan 04 2020
For n > 2, a(n) is of the form 2^e_1 * p_2^e_2 * ... * p_m^e_m, where e_m = 1 and e = floor(log_2(p_m)) <= e_1. Therefore, 2^e * p_m^e_m is a primitive Zumkeler number (A180332). Therefore, 2^e_1 * p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 2, a(n) = 2^e_1 * p_m^e_m * r, where r is relatively prime to 2*p_m, is a Zumkeller number (see my proof at A002182 for details). - Ivan N. Ianakiev, May 10 2020
For n > 1, 2|(a(n)+2) ... n|(a(n)+n), so a(n)+2 .. a(n)+n are all composite and (part of) a prime gap of at least n.  (Compare n!+2 .. n!+n). - Stephen E. Witham, Oct 09 2021

			LCM of {1,2,3,4,5,6} = 60. The primes up to 6 are 2, 3 and 5. floor(log(6)/log(2)) = 2 so the exponent of 2 is 2.
floor(log(6)/log(3)) = 1 so the exponent of 3 is 1.
floor(log(6)/log(5)) = 1 so the exponent of 5 is 1. Therefore, a(6) = 2^2 * 3^1 * 5^1 = 60. - _David A. Corneth_, Jun 02 2017

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 365.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..2308 (first 501 terms from T. D. Noe)
R. Anderson and N. J. A. Sloane, Correspondence, 1975.
Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, The Exponent of a Group: Properties, Computations and Applications, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
Javier Cilleruelo, Juanjo Rué, Paulius Šarka, and Ana Zumalacárregui, The least common multiple of sets of positive integers, arXiv:1112.3013 [math.NT], 2011.
R. E. Crandall and C. Pomerance, Prime numbers: a computational perspective, MR2156291, p. 61.
Roger B. Eggleton, Least Common Multiple of {1,2,...,n}, Mathematics Magazine, 61(1) (1988), pp. 47-48, Problem 1252.
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(9) (2009), 836-839.
Steven Finch, Cilleruelo's LCM Constants, 2013. [Cached copy, with permission of the author]
V. L. Gavrikov, On property of least common multiple to be a D-magic number, arXiv:1806.09264 [math.NT], 2018.
S. Labbé and E. Pelantová, Palindromic sequences generated from marked morphisms, arXiv:1409.7510 [math.CO], 2014.
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (6) (2002) 534-543. arXiv:math/0008177 [math.NT], 2000-2001.
Peter Luschny and S. Wehmeier, The lcm(1, 2, ..., n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Des MacHale and Joseph Manning, Maximal runs of strictly composite integers, The Mathematical Gazette, 99, pp 213-219 (2015).
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384 [math.CA], 2009.
M. Nair, On Chebychev-type inequalities for primes Amer. Math. Monthly 89(2) (1982), 126-129.
S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society ser. 2, vol. XIV, no. 1 (1915), pp 347-409. (A variant of a better quality with an additional footnote is available here.)
E. S. Selmer, On the number of prime divisors of a binomial coefficient, Math. Scand. 39 (1976), no. 2, 271-281 (1977).
Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. AMS 131 (2003), 3335.
Rosemary Sullivan and Neil Watling, Independent divisibility pairs on the set of integers from 1 to n, INTEGERS 13 (2013) #A65.
M. Tchebichef, Mémoire sur les nombres premiers, J. Math. Pures Appliquées 17 (1852), 366-390.
Helge von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1) (1901), 159-182.
Eric Weisstein's World of Mathematics, Least Common Multiple, Chebyshev Functions, Mangoldt Function.
Index to divisibility sequences
Index entries for "core" sequences
Index entries for sequences related to lcm's

Row products of A133233.
Cf. A000142, A000793, A002110, A002182, A002201, A002944, A014963, A020500, A025527, A038610, A051173, A064446, A064859, A069513, A072938, A093880, A094348, A096179, A099996, A102910, A106037, A119682, A179661, A193181, A225558, A225630, A225632, A225640, A225642.
Cf. A025528 (number of prime factors of a(n) with multiplicity).
Cf. A275120 (lengths of runs of consecutive equal terms), A276781 (ordinal transform from term a(1)=1 onward).

a003418 = foldl lcm 1 . enumFromTo 2
-- Reinhard Zumkeller, Apr 04 2012, Apr 25 2011

[1] cat [Exponent(SymmetricGroup(n)) : n in [1..28]]; // Arkadiusz Wesolowski, Sep 10 2013

[Lcm([1..n]): n in [0..30]]; // Bruno Berselli, Feb 06 2015

A003418 := n-> lcm(seq(i,i=1..n));
HalfFarey := proc(n) local a,b,c,d,k,s; a := 0; b := 1; c := 1; d := n; s := NULL; do k := iquo(n + b, d); a, b, c, d := c, d, k*c - a, k*d - b; if 2*a > b then break fi; s := s,(a/b); od: [s] end: LCM := proc(n) local i; (1/2)*mul(2*sin(Pi*i),i=HalfFarey(n))^2 end: # Peter Luschny
# next Maple program:
a:= proc(n) option remember; `if`(n=0, 1, ilcm(n, a(n-1))) end:
seq(a(n), n=0..33);  # Alois P. Heinz, Jun 10 2021

Table[LCM @@ Range[n], {n, 1, 40}] (* Stefan Steinerberger, Apr 01 2006 *)
FoldList[ LCM, 1, Range@ 28]
A003418[0] := 1; A003418[1] := 1; A003418[n_] := A003418[n] = LCM[n,A003418[n-1]]; (* Enrique Pérez Herrero, Jan 08 2011 *)
Table[Product[Prime[i]^Floor[Log[Prime[i], n]], {i, PrimePi[n]}], {n, 0, 28}] (* Wei Zhou, Jun 25 2011 *)
Table[Product[Cyclotomic[n, 1], {n, 2, m}], {m, 0, 28}] (* Fred Daniel Kline, May 22 2014 *)
a1[n_] := 1/12 (Pi^2+3(-1)^n (PolyGamma[1,1+n/2] - PolyGamma[1,(1+n)/2])) // Simplify
a[n_] := Denominator[Sqrt[a1[n]]];
Table[If[IntegerQ[a[n]], a[n], a[n]*(a[n])[[2]]], {n, 0, 28}] (* Gerry Martens, Apr 07 2018 [Corrected by Vaclav Kotesovec, Jul 16 2021] *)

a(n)=local(t); t=n>=0; forprime(p=2,n,t*=p^(log(n)\log(p))); t

a(n)=if(n<1,n==0,1/content(vector(n,k,1/k)))

a(n)=my(v=primes(primepi(n)),k=sqrtint(n),L=log(n+.5));prod(i=1,#v,if(v[i]>k,v[i],v[i]^(L\log(v[i])))) \\ Charles R Greathouse IV, Dec 21 2011

a(n)=lcm(vector(n,i,i)) \\ Bill Allombert, Apr 18 2012 [via Charles R Greathouse IV]

n=1; lim=100; i=1; j=1; until(n==lim, a=lcm(j,i+1); i++; j=a; n++; print(n" "a);); \\ Mike Winkler, Sep 07 2013

from functools import reduce
from operator import mul
from sympy import sieve
def integerlog(n,b): # find largest integer k>=0 such that b^k <= n
    kmin, kmax = 0,1
    while b**kmax <= n:
        kmax *= 2
    while True:
        kmid = (kmax+kmin)//2
        if b**kmid > n:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmin
def A003418(n):
    return reduce(mul,(p**integerlog(n,p) for p in sieve.primerange(1,n+1)),1) # Chai Wah Wu, Mar 13 2021

# generates initial segment of sequence
from math import gcd
from itertools import accumulate
def lcm(a, b): return a * b // gcd(a, b)
def aupton(nn): return [1] + list(accumulate(range(1, nn+1), lcm))
print(aupton(30)) # Michael S. Branicky, Jun 10 2021

[lcm(range(1,n)) for n in range(1, 30)] # Zerinvary Lajos, Jun 06 2009

(define (A003418 n) (let loop ((n n) (m 1)) (if (zero? n) m (loop (- n 1) (lcm m n))))) ;; Antti Karttunen, Jan 03 2018

The prime number theorem implies that lcm(1,2,...,n) = exp(n(1+o(1))) as n -> infinity. In other words, log(lcm(1,2,...,n))/n -> 1 as n -> infinity. - Jonathan Sondow, Jan 17 2005
a(n) = Product (p^(floor(log n/log p))), where p runs through primes not exceeding n (i.e., primes 2 through A007917(n)). - Lekraj Beedassy, Jul 27 2004
Greg Martin showed that a(n) = lcm(1,2,3,...,n) = Product_{i = Farey(n), 0 < i < 1} 2*Pi/Gamma(i)^2. This can be rewritten (for n > 1) as a(n) = (1/2)*(Product_{i = Farey(n), 0 < i <= 1/2} 2*sin(i*Pi))^2. - Peter Luschny, Aug 08 2009
Recursive formula useful for computations: a(0)=1; a(1)=1; a(n)=lcm(n,a(n-1)). - Enrique Pérez Herrero, Jan 08 2011
From Enrique Pérez Herrero, Jun 01 2011: (Start)
a(n)/a(n-1) = A014963(n).
if n is a prime power p^k then a(n)=a(p^k)=p*a(n-1), otherwise a(n)=a(n-1).
a(n) = Product_{k=2..n} (1 + (A007947(k)-1)*floor(1/A001221(k))), for n > 1. (End)
a(n) = A079542(n+1, 2) for n > 1.
a(n) = exp(Sum_{k=1..n} Sum_{d|k} moebius(d)*log(k/d)). - Peter Luschny, Sep 01 2012
a(n) = A025529(n) - A027457(n). - Eric Desbiaux, Mar 14 2013
a(n) = exp(Psi(n)) = 2 * Product_{k=2..A002088(n)} (1 - exp(2*Pi*i * A038566(k+1) / A038567(k))), where i is the imaginary unit, and Psi the second Chebyshev's function. - Eric Desbiaux, Aug 13 2014
a(n) = A064446(n)*A038610(n). - Anthony Browne, Jun 16 2016
a(n) = A000142(n) / A025527(n) = A000793(n) * A225558(n). - Antti Karttunen, Jun 02 2017
log(a(n)) = Sum_{k>=1} (A309229(n, k)/k - 1/k). - Mats Granvik, Aug 10 2019
From Petros Hadjicostas, Jul 24 2020: (Start)
Nair (1982) proved that 2^n <= a(n) <= 4^n for n >= 9. See also Farhi (2009). Nair also proved that
a(n) = lcm(m*binomial(n,m): 1 <= m <= n) and
a(n) = gcd(a(m)*binomial(n,m): n/2 <= m <= n). (End)
Sum_{n>=1} 1/a(n) = A064859. - Bernard Schott, Aug 24 2020

A188666 Largest m <= n such that lcm(m, m+1, ..., n) = lcm(1, 2, ..., n).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 7, 7, 7, 7, 8, 8, 9, 9, 11, 11, 11, 11, 13, 13, 13, 13, 16, 16, 16, 16, 16, 16, 17, 17, 19, 19, 19, 19, 23, 23, 23, 23, 23, 23, 23, 23, 25, 25, 25, 25, 27, 27, 27, 27, 29, 29, 29, 29, 31, 31, 31, 31, 32, 32, 37, 37, 37, 37, 37, 37

Offset: 1

Reinhard Zumkeller, Apr 25 2011

nonn

By definition: A003418(n) = lcm(a(n), a(n)+1, ... n)
and lcm(m, m+1, ... n) < A003418(n) for m > a(n);
all terms are prime powers, cf. A000961: A010055(a(n)) = 1;
a(A110654(n)) = A000015(n);
floor(n/2)+1 <= a(n) < a(2*(a(n));
A000961(n+1) = a(2*A000961(n)) = a(A138929(n)), cf. formula.
A237709 gives number of occurrences of n-th prime power. - Reinhard Zumkeller, Feb 12 2014
Conjecture: a(n) = A000015(floor(n/2)+1). - Georg Fischer, Nov 29 2022

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Least Common Multiple
Wikipedia, Least Common Multiple
Index entries for sequences related to lcm's

Cf. A000015, A051173, A099996.

import Data.List (elemIndices)
a188666 n = a188666_list !! (n-1)
a188666_list = g 1 a000961_list where
   g n pps'@(pp:pp':pps) | n < 2*pp  = pp  : g (n+1) pps'
                         | otherwise = pp' : g (n+1) (pp':pps)
-- Alternative, rewriting the definition, but less efficient:
a188666' n = last $ elemIndices (f 1) $ map f [0..n] where
   f from = foldl lcm 1 [from..n]

Table[Block[{k = n, m = LCM @@ Range[n]},  While[LCM @@ Range[k, n] != m, k--]; k], {n, 69}] (* Michael De Vlieger, Nov 29 2022 *)

A188666(n)=L=lcm(n=vector(n-1,k,k+1));!for(m=1,#n,lcm(n[-m..-1])==L&&return(#n+2-m))\\ Rather illustrative than efficient. - M. F. Hasler, Jul 25 2015

from itertools import count
from sympy import factorint
def A188666(n): return next(filter(lambda m:len(factorint(m))<=1, count((n>>1)+1))) # Chai Wah Wu, Oct 25 2024

a(n) = A000961(k+1) for n: 2*A000961(k) <= n < 2*A000961(k+1), k > 0.

A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^ic(n,i)binomial(i*p,p) is divisible by p^(2n-1).

Original entry on oeis.org

1, 2, 1, 12, 9, 2, 60, 54, 20, 3, 840, 840, 400, 105, 12, 2520, 2700, 1500, 525, 108, 10, 27720, 31185, 19250, 8085, 2268, 385, 30, 360360, 420420, 280280, 133770, 45864, 10780, 1560, 105, 720720, 864864, 611520, 321048, 127008, 36960, 7488, 945, 56, 12252240, 15036840, 11138400, 6297480, 2776032, 942480

Offset: 1

Max Alekseyev, Feb 06 2016

			n=1: 1
n=2: 2, 1
n=3: 12, 9, 2
n=4: 60, 54, 20, 3
n=5: 840, 840, 400, 105, 12
...
For all primes p>3, p^3 divides 2 - binomial(2*p,p) (cf. A087754).
For all primes p>5, p^5 divides 12 - 9*binomial(2*p,p) + 2*binomial(3*p,p) (cf. A268589).
For all primes p>7, p^7 divides 60 - 54*binomial(2*p,p) + 20*binomial(3*p,p) - 3*binomial(4*p,p) (cf. A268590).

R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0; also arXiv, arXiv:1602.02632 [math.NT], 2016-2018.

Cf. A099996 (first column), A068550 (diagonal), A087754, A268589, A268590, A254593.

a3418[n_] := LCM @@ Range[n];
c[1, 1] = 1; c[n_, i_] := a3418[2(n-1)] Binomial[2n-1, n-i] ((2i-1)/i/ Binomial[2n-1, n]);
Table[c[n, i], {n, 1, 10}, {i, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2018 *)

{ A268512(n,i) = lcm(vector(2*(n-1),i,i)) * binomial(2*n-1,n-i) * (2*i-1) / i / binomial(2*n-1,n) }

c(n,i) = A003418(2*(n-1))*binomial(2*n-1,n-i)*(2*i-1)/i/binomial(2*n-1,n).

A068550 a(n) = lcm{1, ..., 2n} / binomial(2n, n).

Original entry on oeis.org

1, 1, 2, 3, 12, 10, 30, 105, 56, 252, 1260, 330, 1980, 2574, 2002, 15015, 240240, 61880, 15912, 151164, 38760, 406980, 4476780, 1144066, 13728792, 24515700, 6249100, 84362850, 21474180, 5462730, 81940950, 1270084725, 645122400

Offset: 0

N. J. A. Sloane, Mar 23 2002

Known to be always an integer.

Hojoo Lee, Re: LCM [1,2,..,N] > 2^{N-1}, NMBRTHRY Mailing List, Feb 18 2002.
Daniel Ropp, Solution to Problem 31, Crux Mathematicorum, Vol. 13, No. 10 (1987), p. 314.

Bisection of A180000 and A263673.

a[0] = 1; a[n_] := (LCM @@ Range[2*n])/Binomial[2*n, n]; Array[a, 33, 0] (* Amiram Eldar, Mar 06 2022 *)

a(n) = lcm([1..2*n])/binomial(2*n, n); \\ Michel Marcus, Mar 06 2022

a(n) = A099996(n) / A000984(n) = A003418(2*n) / A001405(2*n) = A180000(2*n) = A263673(2*n).

a(n) = n * A068553(n) = n * A048619(2*n-1).

a(0)=1 prepended by Max Alekseyev, Oct 23 2015

A093880 a(n) = lcm(1, 2, ..., 2n) / lcm(1, 2, ..., n).

Original entry on oeis.org

2, 6, 10, 70, 42, 462, 858, 858, 4862, 92378, 8398, 193154, 74290, 222870, 6463230, 200360130, 11785890, 11785890, 22951470, 22951470, 941010270, 40463441610, 1759280070, 82686163290, 115760628606, 115760628606, 2045104438706

Offset: 1

Amarnath Murthy, Apr 22 2004

nonn

Also, lcm(n+1, n+2, ..., 2n-1, 2n) / lcm(1, 2, ..., n-1, n).

			The LCM of {1,2,3,4,5,6} is 60 and the LCM of {1,2,3} is 6, so a(3) = 60/6 = 10.

T. D. Noe, Table of n, a(n) for n = 1..500
J. Sondow, Criteria for irrationality of Euler's constant, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003) 3335-3344.
Eric Weisstein's World of Mathematics, Least Common Multiple
Index entries for sequences related to lcm's

Cf. A080397.

a:=n->lcm(seq(j,j=n+1..2*n))/lcm(seq(j,j=1..n)): seq(a(n),n=1..32); # Emeric Deutsch, Feb 02 2006

f[n_] := LCM @@ Table[i, {i, 2n}]/LCM @@ Table[i, {i, n}]; Table[ f[n], {n, 27}] (* Robert G. Wilson v, Jan 22 2005 *)

The prime number theorem implies that a(n) = e^(n(1+o(1))) as n -> infinity. In other words, log(a(n))/n -> 1 as n -> infinity. - Jonathan Sondow, Jan 17 2005

a(n) = A003418(2n)/A003418(n) = A099996(n)/A003418(n).

More terms from Emeric Deutsch, Feb 02 2006

Entry revised by N. J. A. Sloane, Jan 24 2007

A263673 a(n) = lcm{1,2,...,n} / binomial(n,floor(n/2)).

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 3, 12, 12, 20, 10, 60, 30, 210, 105, 56, 56, 504, 252, 2520, 1260, 660, 330, 3960, 1980, 5148, 2574, 4004, 2002, 30030, 15015, 240240, 240240, 123760, 61880, 31824, 15912, 302328, 151164, 77520, 38760, 813960, 406980, 8953560, 4476780, 2288132, 1144066, 27457584, 13728792, 49031400

Offset: 0

Max Alekseyev, Oct 23 2015

From Robert Israel, Oct 23 2015: (Start)
If n = 2^k, a(n) = a(n-1).
If n = p^k where p is an odd prime and k >= 1, 2*n*a(n) = p*(n+1)*a(n-1).
If n is even and not a prime power, 2*a(n) = a(n-1).
If n is odd and not a prime power, 2*n*a(n) = (n+1)*a(n-1). (End)

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A068550, A180000.

a := n -> lcm(seq(k,k=1..n))/binomial(n,iquo(n,2)):
seq(a(n), n=0..49); # Peter Luschny, Oct 23 2015

Join[{1}, Table[LCM @@ Range[n]/Binomial[n, Floor[n/2]], {n, 1, 50}]] (* or *) Table[Product[Cyclotomic[k, 1], {k, 2, n}]/Binomial[n, Floor[n/2]], {n, 0, 50}] (* G. C. Greubel, Apr 17 2017 *)

A263673(n) = lcm(vector(n,i,i)) / binomial(n,n\2);

a(n) = A003418(n) / A001405(n).
a(n) = A048619(n-1) * A110654(n).
a(2*n) = A068550(n) = A099996(n) / A000984(n).
a(n) = A180000(n)*A152271(n). - Peter Luschny, Oct 23 2015
a(n) = (e/2)^(n + o(1)). - Charles R Greathouse IV, Oct 23 2015

A120114 a(n) = lcm(1, ..., 2n+4)/lcm(1, ..., 2n+2).

Original entry on oeis.org

6, 5, 14, 3, 11, 13, 2, 17, 19, 1, 23, 5, 3, 29, 62, 1, 1, 37, 1, 41, 43, 1, 47, 7, 1, 53, 1, 1, 59, 61, 2, 1, 67, 1, 71, 73, 1, 1, 79, 3, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103, 1, 107, 109, 1, 113, 1, 1, 1, 11, 1, 5, 254, 1, 131, 1, 1, 137, 139, 1, 1, 1, 1

Offset: 0

Paul Barry, Jun 09 2006

The subdiagonal of A120113 is -a(n).
From Robert Israel, Dec 03 2024: (Start)
a(n) is the product of the primes p such that 2*n + 3 or 2*n + 4 is a power of p.
Thus: a(n) = 1 if and only if neither 2*n + 3 nor 2*n + 4 is in A000961.
      if n + 1 = 2^k - 1 is a Mersenne number but not a Mersenne prime, then a(n) = 2;
      if n + 1 = 2^k - 1 is a Mersenne prime, then a(n) = 2 * (2^k - 1);
      otherwise a(n) is odd. (End)
Conjectures from Davide Rotondo, Dec 02 2024: (Start)
Except for 2, if a(n) is even then a(n)/2 is a Mersenne prime.
If a(n)=1 or a(n)=2 then (n*2)+3 is in A061346, or also, or (n+1) is in A083390. (End)

Muniru A Asiru, Table of n, a(n) for n = 0..5000

Cf. A000961, A099996, A120113.

List([0..75],n->Lcm(List([1..2*n+4],i->i))/Lcm(List([1..2*n+2],i->i))); # Muniru A Asiru, Mar 04 2019

A120114:= func< n | Lcm([1..2*n+4])/Lcm([1..2*n+2]) >;
[A120114(n): n in [0..100]]; // G. C. Greubel, May 05 2023

f:= proc(n) local t,x,S;
   t:= 1;
   for x from 2*n+3 to 2*n+4 do
     S:= numtheory:-factorset(x);
     if nops(S) = 1 then t:= t*S[1] fi;
   od;
   t
end proc:
map(f, [$0..100]); # Robert Israel, Dec 03 2024

Table[(LCM@@Range[2n+4])/LCM@@Range[2n+2],{n,0,100}] (* Harvey P. Dale, Dec 15 2017 *)

def A120114(n):
    return lcm(range(1,2*n+5)) // lcm(range(1,2*n+3))
[A120114(n) for n in range(101)] # G. C. Greubel, May 05 2023

a(n) = A099996(n+2)/A099996(n+1). - Michel Marcus, May 06 2023

More terms from Harvey P. Dale, Dec 15 2017

A025540 Least common multiple of {C(0,0), C(2,1), ..., C(2n,n)}.

Original entry on oeis.org

1, 2, 6, 60, 420, 1260, 13860, 360360, 360360, 6126120, 116396280, 116396280, 2677114440, 13385572200, 40156716600, 2329089562800, 72201776446800, 72201776446800, 72201776446800, 2671465728531600, 2671465728531600

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mathoverflow, Asymptotic of LCM(binomial(2k,k)), 1<=k<=n

Cf. A000984.

Table[LCM@@Table[Binomial[2n,n],{n,0,i}],{i,0,30}] (* Harvey P. Dale, Jun 04 2012 *)

a(n) = lcm(vector(n, k, binomial(2*k, k))); \\ Michel Marcus, Apr 13 2016

a(n) = 2^floor(log(n+1)/log(2)) * Prod( p^floor(log(2n)/log(p)) ), where the product is taken over all odd primes p below 2n. - Max Alekseyev, Apr 13 2016

If n = 2^k - 1, then a(n) = A099996(n) = A003418(2*n); otherwise a(n) = A099996(n)/2 = A003418(2*n)/2. - Max Alekseyev, Apr 13 2016

A156744 Coefficients for estimation of derivative from equally spaced numerical data using the Lagrange interpolating polynomial.

Original entry on oeis.org

-1, 0, 1, 1, -8, 0, 8, -1, -1, 9, -45, 0, 45, -9, 1, 3, -32, 168, -672, 0, 672, -168, 32, -3

Offset: 1

Bruce Boghosian, Feb 14 2009

An effective way to approximate the derivative of equally spaced numerical data is to differentiate its Lagrange interpolating polynomial. If y[x] is equally spaced data from x = -n to +n, its Lagrange interpolating polynomial P(x) has degree 2*n+1. Then P'(0) may be expressed as a weighted sum over the y[x]. This is the triangle of coefficients C[n,m] such that P'(0) = (1/d[n]) * Sum_{m=-n}^n C[n,m] y[m]. The denominator d[n] is given by sequence A099996. This is very useful in numerical analysis. For example, when n=1, this gives the centered difference approximation to the derivative.

			Irregular triangle begins:
  -1, 0, 1;
  1, -8, 0, 8, -1;
  -1, 9, -45, 0, 45, -9, 1;
  3, -32, 168, -672, 0, 672, -168, 32, -3;
  -2, 25, -150, 600, -2100, 0, 2100, -600, 150, -25, 2;  ...

When divided by sequence A099996, this triangle gives the coefficients needed to estimate derivatives from equally spaced numerical data using Lagrange interpolation. The first and last entry of each row of the triangle has absolute value lcm{1, 2, ..., 2*n}/n*binomial(2n, n), as seen in A068553.

facs[x_, j_, n_] := Product[If[k == j, 1, x - k], {k, -n, n}]
coefs[x_, n_] := Table[facs[x, j, n]/facs[j, j, n], {j, -n, n}]
d[n_] := Apply[LCM, Table[j, {j, 1, 2*n}]]
dCoefs[x_, n_] := d[n] D[coefs[y, n], y] /. {y -> x}
MatrixForm[Table[dCoefs[0, n], {n, 1, 5}]]

cp's OEIS Frontend

A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.

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A188666 Largest m <= n such that lcm(m, m+1, ..., n) = lcm(1, 2, ..., n).

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A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^i*c(n,i)*binomial(i*p,p) is divisible by p^(2n-1).

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A068550 a(n) = lcm{1, ..., 2n} / binomial(2n, n).

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A093880 a(n) = lcm(1, 2, ..., 2n) / lcm(1, 2, ..., n).

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A263673 a(n) = lcm{1,2,...,n} / binomial(n,floor(n/2)).

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A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^ic(n,i)binomial(i*p,p) is divisible by p^(2n-1).