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A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.
(Formerly M1590)
241
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Product over all primes of highest power of prime less than or equal to n. a(0) = 1 by convention.

Also smallest number such that its 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. (.. with the constraint n>=3).

Also the minimal exponent of the symmetric group S_n (i.e. the least positive integer a(n) for which x^a(n)=1 for all x in S_n). - Franz Vrabec, Dec 28 2008

Periods of the sequences b(n)=Sum{i=0..k-1}{(n+i} mod (k-i)} for k=0,1,2,3,... - Paolo P. Lava, Feb 18 2009

Corollary 3 of Farhi gives a simple proof that A003418(n) => 2^(n-1). The main theorem proved in Farhi is the identity lcm{binom{k,0}, binom(k,1), ..., binom(k,k) = lcm(1, 2, ..., k, k + 1)/(k + 1) for all k in N. - Jonathan Vos Post, Jun 15 2009

a[x]=exp(psi(x)) where psi(x)=log(lcm(1,2,...,floor(x))) is the Chebyshev function of the second kind. - Stephen Crowley, Jul 04 2009

Appears to be row products of the triangle T(n,k) = b(A010766) where b = A130087/A130086. - Mats Granvik, Jul 08 2009

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+1)/a(n). - Asher Auel (asher.auel(AT)reed.edu)

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(A003418(n)*2*i*Pi/k) ). - Eric Desbiaux, Sep 13 2012

First column minus second column of A027446. - Eric Desbiaux, Mar 29 2013.

a(n)>0 is the smallest number k such that n is the n-th divisor of k. - Michel Lagneau, Apr 24 2014

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..500

Javier Cilleruelo, Juanjo Rué, Paulius Šarka, and Ana Zumalacárregui, The least common multiple of sets of positive integers (2011).

R. E. Crandall, C. Pomerance, Prime numbers: a computational perspective, MR2156291, page 61

Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, arXiv:0906.2295.

Steven Finch, Cilleruelo's LCM Constants, 2013.

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (6) (2002) 534. arXiv:math.NT/0008177

Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384

E. S. Selmer, On the number of prime divisors of a binomial coefficient Math. Scand. 39 (1976), no. 2, 271-281 (1977).

J. 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, Memoire sur les nombres premiers, J. Math. Pures Appliquees 17 (1852) 366.

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

D. Williams, LCM

Index to divisibility sequences

Index entries for "core" sequences

Index entries for sequences related to lcm's

FORMULA

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} = Prod_{i=Farey(n),0<i<1} 2Pi/Gamma(i)^2. This can be rewritten (for n>1) as a(n) = (1/2)[Prod_{ i=Farey(n),0<i<=1/2} 2sin(iPI)]^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) = prod(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

EXAMPLE

LCM of {1,2,3,4,5,6} = 60.

MAPLE

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

MATHEMATICA

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 *)

PROG

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

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

(PARI) 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

(PARI) a(n)=lcm(vector(n, i, i)) \\ Bill Allombert, Apr 18 2012

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

(Haskell)

a003418 = foldl lcm 1 . enumFromTo 2

-- Reinhard Zumkeller, Apr 04 2012, Apr 25 2011

(PARI) {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

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

CROSSREFS

Row products of A133233.

Cf. A002944, A102910, A093880, A133233, A099996, A051173, A014963, A069513, A096179, A179661, A094348, A002182, A002201, A072938, A106037, A002110.

Cf. A025527, A225558, A225630, A225632, A225640, A225642.

Sequence in context: A085911 A211418 A058312 * A109935 A065887 A072181

Adjacent sequences:  A003415 A003416 A003417 * A003419 A003420 A003421

KEYWORD

nonn,easy,core,nice

AUTHOR

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

STATUS

approved

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Last modified July 30 05:27 EDT 2014. Contains 245052 sequences.