Search: id:A003418
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%I A003418 M1590
%S A003418 1,1,2,6,12,60,60,420,840,2520,2520,27720,27720,360360,360360,360360,
%T A003418 720720,12252240,12252240,232792560,232792560,232792560,232792560,
%U A003418 5354228880,5354228880,26771144400,26771144400,80313433200,80313433200
%N A003418 a(0) = 1; for n >= 1, a(n) = least common multiple (or lcm) of {1, 2, 
               ..., n}.
%C A003418 Product over all primes of highest power of prime less than or equal 
               to n. a(0) = 1 by convention.
%C A003418 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
%C A003418 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
%C A003418 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]
%C A003418 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]
%C A003418 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]
%C A003418 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]
%C A003418 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]
%D A003418 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A003418 J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 365.
%D A003418 Selmer, Ernst S.; On the number of prime divisors of a binomial coefficient. 
               Math. Scand. 39 (1976), no. 2, 271-281 (1977).
%D A003418 J. Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. 
               Math. Soc. 131 (2003) 3335-3344.
%H A003418 T. D. Noe, <a href="b003418.txt">Table of n, a(n) for n = 0..500</a>
%H A003418 Bakir Farhi, <a href="http://arxiv.org/abs/0906.2295">An identity involving 
               the least common multiple of binomial coefficients and its application</
               a>, American Mathematical Monthly (November 2009). [From Jonathan 
               Vos Post (jvospost3(AT)gmail.com), Jun 15 2009]
%H A003418 Greg Martin, <a href="http://arxiv.org/abs/0907.4384">A product of Gamma 
               function values at fractions with the same denominator</a>, Jul 24, 
               2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 28 2009]
%H A003418 J. Sondow, <a href="http://www.ams.org/journal-getitem?pii=S0002-9939-03-07081-3">
               Criteria for irrationality of Euler's constant</a>
%H A003418 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LeastCommonMultiple.html">Least Common Multiple</a>
%H A003418 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               ChebyshevFunctions.html">Link to a section of The World of Mathematics.</
               a>
%H A003418 D. Williams, <a href="http://www.louisville.edu/~dawill03/crypto/LCM.html">
               LCM</a>
%H A003418 <a href="Sindx_Lc.html#lcm">Index entries for sequences related to lcm's</
               a>
%H A003418 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%H A003418 <a href="http://mathworld.wolfram.com/ChebyshevFunctions.html">Chebyshev 
               Functions at Mathworld</a> [From Stephen Crowley (crow(AT)crowlogic.net), 
               Jul 04 2009]
%F A003418 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
%F A003418 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
%F A003418 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.
%e A003418 LCM of {1,2,3,4,5,6} = 60.
%p A003418 A003418 := n-> lcm(seq(i,i=1..n));
%p A003418 seq (denom(sum((-1)^i/i, i = 1..n)), n=0..30); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               May 14 2008
%p A003418 [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:
%t A003418 Table[LCM @@ Range[n], {n, 1, 40}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 01 2006
%o A003418 (PARI) a(n)=local(t); t=n>=0; forprime(p=2,n,t*=p^(log(n)\log(p))); t
%o A003418 (PARI) a(n)=if(n<1,n==0,1/content(vector(n,k,1/k)))
%o A003418 (Other) sage: [lcm(range(1,n)) for n in xrange(1, 30)] # [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]
%Y A003418 Row products of A133233. - Mats Granvik (mgranvik(AT)abo.fi), Jan 22 
               2008
%Y A003418 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]
%Y A003418 Cf. A002944, A102910, A093880, A133233.
%Y A003418 a(n) = A079542(n+1, 2) for n>1.
%Y A003418 It appears that A020500(n) = A003418(n+1)/a(n) - Asher Auel (asher.auel(AT)reed.edu)
%Y A003418 Cf. A094348, A002182, A002201, A072938, A106037, A002110.
%Y A003418 Sequence in context: A083268 A085911 A058312 this_sequence A109935 A065887 
               A072181
%Y A003418 Adjacent sequences: A003415 A003416 A003417 this_sequence A003419 A003420 
               A003421
%K A003418 nonn,easy,core,nice,new
%O A003418 0,3
%A A003418 Roland Anderson (roland.anderson(AT)swipnet.se)
%E A003418 Updated a reference, - Charles R Greathouse IV (charles.greathouse(AT)case.edu), 
               Oct 28 2009

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