This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1. (Formerly M1590) 295
 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 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 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. (This is wrong for n=1 and n=2. Should "for n large enough" be added? - Georgi Guninski, Oct 22 2011) 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(binomial(k,0), binomial(k,1), ..., binomial(k,k)) = lcm(1, 2, ..., k, k + 1)/(k + 1) for all k in N. - 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+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 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 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 and Seiichi Manyama, Table of n, a(n) for n = 0..2308 (first 501 terms from T. D. Noe) R. Anderson & N. J. A. Sloane, Correspondence, 1975 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, 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 [math.NT], 2009. 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, E. Pelantová, Palindromic sequences generated from marked morphisms, arXiv preprint arXiv:1409.7510 [math.CO], 2014. J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (6) (2002) 534. arXiv:math.NT/0008177 P. 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. 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, Mémoire sur les nombres premiers, J. Math. Pures Appliquées 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 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) = Product_{i=Farey(n), 0 1) as a(n) = (1/2)*(Product_{i=Farey(n), 0 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 EXAMPLE 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 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 := 1; A003418 := 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_] := 1/432 (Pi^2+3(-1)^n (PolyGamma[1, 1+n/2] - PolyGamma[1, (1+n)/2])) // Simplify a[n_] := Denominator[6 Sqrt[a1[n]]] Table[a[n], {n, 0, 28}] (* Gerry Martens, April 07 2018 *) 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 [via Charles R Greathouse IV] (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 (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 (MAGMA)  cat [Exponent(SymmetricGroup(n)) : n in [1..28]]; // Arkadiusz Wesolowski, Sep 10 2013 (MAGMA) [Lcm([1..n]): n in [0..30]]; // Bruno Berselli, Feb 06 2015 (Scheme) (define (A003418 n) (let loop ((n n) (m 1)) (if (zero? n) m (loop (- n 1) (lcm m n))))) ;; Antti Karttunen, Jan 03 2018 CROSSREFS Row products of A133233. Cf. A000142, A000793, A002944, A102910, A093880, A099996, A051173, A014963, A069513, A096179, A179661, A094348, A002182, A002201, A072938, A106037, A002110, A025527, A225558, A225630, A225632, A225640, A225642, A038610, A064446, A193181, A119682. 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 21 15:26 EDT 2019. Contains 321370 sequences. (Running on oeis4.)