0,3

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 (reinhard.zumkeller(AT)gmail.com), Dec 09 2002

An assertion equivalent to the Riemann hypothesis is: | ln(a(n)) - n | < sqrt(n)*(ln(n))^2. - Lekraj Beedassy (blekraj(AT)yahoo.com), Aug 27 2006

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). [From Franz Vrabec (franz.vrabec(AT)planetuniqa.at), 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,... [From Paolo P. Lava (ppl(AT)spl.at), 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. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 15 2009]

a[x]=exp(psi(x)) where psi(x)=ln(lcm(1,2,...,floor(x))) is the Chebyshev function of the second kind. [From Stephen Crowley (crow(AT)crowlogic.net), Jul 04 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). [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 28 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 365.

Selmer, Ernst S.; 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. Amer. Math. Soc. 131 (2003) 3335-3344.

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

Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, American Mathematical Monthly (November 2009). [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 15 2009]

Greg Martin, A product of Gamma function values at fractions with the same denominator, Jul 24, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 28 2009]

J. Sondow, Criteria for irrationality of Euler's constant

Eric Weisstein's World of Mathematics, Least Common Multiple

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

D. Williams, LCM

Index entries for sequences related to lcm's

Index entries for "core" sequences

Chebyshev Functions at Mathworld [From Stephen Crowley (crow(AT)crowlogic.net), Jul 04 2009]

The prime number theorem implies that LCM(1,2,...,n) = e^(n(1+o(1))) as n -> infinity. In other words, ln(LCM(1,2,...,n))/n -> 1 as n -> infinity. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), 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 (blekraj(AT)yahoo.com), Jul 27 2004

Comment from Peter Luschny, Aug 08 2009: 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.

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

A003418 := n-> lcm(seq(i, i=1..n));

seq (denom(sum((-1)^i/i, i = 1..n)), n=0..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 14 2008

[From Peter Luschny] 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:

Table[LCM @@ Range[n], {n, 1, 40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 01 2006

(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)))

(Other) sage: [lcm(range(1, n)) for n in xrange(1, 30)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]

Row products of A133233. - Mats Granvik (mgranvik(AT)abo.fi), Jan 22 2008

Appears to be row products of the triangle T(n,k) = b(A010766) where b = A130087/A130086. [From Mats Granvik (mats.granvik(AT)abo.fi), Jul 08 2009]

Cf. A002944, A102910, A093880, A133233.

a(n) = A079542(n+1, 2) for n>1.

It appears that A020500(n) = A003418(n+1)/a(n) - Asher Auel (asher.auel(AT)reed.edu)

Cf. A094348, A002182, A002201, A072938, A106037, A002110.

Sequence in context: A083268 A085911 A058312 this_sequence A109935 A065887 A072181

Adjacent sequences: A003415 A003416 A003417 this_sequence A003419 A003420 A003421

nonn,easy,core,nice,new

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

Updated a reference, - Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 28 2009