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 A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1. (Formerly M1590) 341
 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, 2329089562800, 2329089562800 (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 whose 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 What is the largest number of consecutive terms that are all equal? I found 112 equal terms from a(370261) to a(370372). - Dmitry Kamenetsky, May 05 2019 Answer: there exist arbitrarily long sequences of consecutive terms with the same value; also, the maximal run of consecutive terms with different values is 5 from a(1) to a(5) (see link Roger B. Eggleton). - Bernard Schott, Aug 07 2019 Related to the inequality (54) in Ramanujan's paper about highly composite numbers A002182, also used in A199337: a(A329570(m))^2 is a (not minimal) bound above which all highly composite numbers are divisible by m, according to the right part of that inequality. - M. F. Hasler, Jan 04 2020 For n > 2, a(n) is of the form 2^e_1 * p_2^e_2 * ... * p_m^e_m, where e_m = 1 and e = floor(log_2(p_m)) <= e_1. Therefore, 2^e * p_m^e_m is a primitive Zumkeler number (A180332). Therefore, 2^e_1 * p_m^e_m is a Zumkeller number (A083207). Therefore, for n > 2, a(n) = 2^e_1 * p_m^e_m * r, where r is relatively prime to 2*p_m, is a Zumkeller number (see my proof at A002182 for details). - Ivan N. Ianakiev, May 10 2020 For n > 1, 2|(a(n)+2) ... n|(a(n)+n), so a(n)+2 .. a(n)+n are all composite and (part of) a prime gap of at least n.  (Compare n!+2 .. n!+n). - Stephen E. Witham, Oct 09 2021 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 and N. J. A. Sloane, Correspondence, 1975. Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, The Exponent of a Group: Properties, Computations and Applications, Disc. Math. and Applications, Springer, Cham (2020), 57-108. 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 and C. Pomerance, Prime numbers: a computational perspective, MR2156291, p. 61. Roger B. Eggleton, Least Common Multiple of {1,2,...,n}, Mathematics Magazine, 61(1) (1988), pp. 47-48, Problem 1252. Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, arXiv:0906.2295 [math.NT], 2009. Bakir Farhi, An identity involving the least common multiple of binomial coefficients and its application, Amer. Math. Monthly 116(9) (2009), 836-839. 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 and E. Pelantová, Palindromic sequences generated from marked morphisms, arXiv:1409.7510 [math.CO], 2014. J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (6) (2002) 534-543. arXiv:math/0008177 [math.NT], 2000-2001. 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. M. Nair, On Chebychev-type inequalities for primes Amer. Math. Monthly 89(2) (1982), 126-129. S. Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society ser. 2, vol. XIV, no. 1 (1915), pp 347-409. (A variant of a better quality with an additional footnote is available here.) E. S. Selmer, On the number of prime divisors of a binomial coefficient, Math. Scand. 39 (1976), no. 2, 271-281 (1977). Jonathan 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), 366-390. 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. 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 < i < 1} 2*Pi/Gamma(i)^2. This can be rewritten (for n > 1) as a(n) = (1/2)*(Product_{i = Farey(n), 0 < i <= 1/2} 2*sin(i*Pi))^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) = Product_{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 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 log(a(n)) = Sum_{k>=1} (A309229(n, k)/k - 1/k). - Mats Granvik, Aug 10 2019 From Petros Hadjicostas, Jul 24 2020: (Start) Nair (1982) proved that 2^n <= a(n) <= 4^n for n >= 9. See also Farhi (2009). Nair also proved that a(n) = lcm(m*binomial(n,m): 1 <= m <= n) and a(n) = gcd(a(m)*binomial(n,m): n/2 <= m <= n). (End) Sum_{n>=1} 1/a(n) = A064859. - Bernard Schott, Aug 24 2020 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 # next Maple program: a:= proc(n) option remember; `if`(n=0, 1, ilcm(n, a(n-1))) end: seq(a(n), n=0..33);  # Alois P. Heinz, Jun 10 2021 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_] := 1/12 (Pi^2+3(-1)^n (PolyGamma[1, 1+n/2] - PolyGamma[1, (1+n)/2])) // Simplify a[n_] := Denominator[Sqrt[a1[n]]]; Table[If[IntegerQ[a[n]], a[n], a[n]*(a[n])[]], {n, 0, 28}] (* Gerry Martens, Apr 07 2018 [Corrected by Vaclav Kotesovec, Jul 16 2021] *) 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 range(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 (Python) from functools import reduce from operator import mul from sympy import sieve def integerlog(n, b): # find largest integer k>=0 such that b^k <= n     kmin, kmax = 0, 1     while b**kmax <= n:         kmax *= 2     while True:         kmid = (kmax+kmin)//2         if b**kmid > n:             kmax = kmid         else:             kmin = kmid         if kmax-kmin <= 1:             break     return kmin def A003418(n):     return reduce(mul, (p**integerlog(n, p) for p in sieve.primerange(1, n+1)), 1) # Chai Wah Wu, Mar 13 2021 (Python) # generates initial segment of sequence from math import gcd from itertools import accumulate def lcm(a, b): return a * b // gcd(a, b) def aupton(nn): return  + list(accumulate(range(1, nn+1), lcm)) print(aupton(30)) # Michael S. Branicky, Jun 10 2021 CROSSREFS Row products of A133233. Cf. A000142, A000793, A002110, A002182, A002201, A002944, A014963, A020500, A025527, A038610, A051173, A064446, A064859, A069513, A072938, A093880, A094348, A096179, A099996, A102910, A106037, A119682, A179661, A193181, A225558, A225630, A225632, A225640, A225642. Cf. A025528 (number of prime factors of a(n) with multiplicity). Cf. A275120 (lengths of runs of consecutive equal terms), A276781 (ordinal transform from term a(1)=1 onward). Sequence in context: A085911 A211418 A058312 * A109935 A347304 A065887 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 September 26 00:53 EDT 2022. Contains 356986 sequences. (Running on oeis4.)