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A003418
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a(0) = 1; for n >= 1, a(n) = least common multiple (or LCM) of {1, 2, ..., n}.
(Formerly M1590)
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220
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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
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OFFSET
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0,3
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COMMENTS
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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). [From 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,... [From 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. [From 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. [From Stephen Crowley (crow(AT)crowlogic.net), Jul 04 2009]
Appears to be row products of the triangle T(n,k) = b(A010766) where b = A130087/A130086. [From 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). [From 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) = A003418(n+1)/a(n) - Asher Auel (asher.auel(AT)reed.edu) n-th distinct value = A051451(n). [From 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.
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REFERENCES
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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).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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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.
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
J. Sondow, Criteria for irrationality of Euler's constant, Proc. AMS 131 (2003) 3335.
Helge von Koch, Sur la distribution des nombres premiers, Acta Math. 24 (1) (1901) 159-182
M. Tchebichef, Memoire sur les nombres premiers, J. Math. Pures Appliquees 17 (1852) 366
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
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FORMULA
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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) = LCL{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
Contribution 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.
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EXAMPLE
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LCM of {1,2,3,4,5,6} = 60.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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)] # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]
(Haskell)
a003418 = foldl lcm 1 . enumFromTo 2
-- Reinhard Zumkeller, Apr 04 2012, Apr 25 2011
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CROSSREFS
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Row products of A133233. - Mats Granvik, Jan 22 2008
Cf. A002944, A102910, A093880, A133233, A099996, A051173, A014963, A069513, A096179, A179661, A094348, A002182, A002201, A072938, A106037, A002110.
Sequence in context: A085911 A211418 A058312 * A109935 A065887 A072181
Adjacent sequences: A003415 A003416 A003417 * A003419 A003420 A003421
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KEYWORD
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nonn,easy,core,nice
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AUTHOR
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Roland Anderson (roland.anderson(AT)swipnet.se)
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STATUS
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approved
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