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%I M1590 #461 Apr 10 2024 14:28:22
%S 1,1,2,6,12,60,60,420,840,2520,2520,27720,27720,360360,360360,360360,
%T 720720,12252240,12252240,232792560,232792560,232792560,232792560,
%U 5354228880,5354228880,26771144400,26771144400,80313433200,80313433200,2329089562800,2329089562800
%N Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.
%C 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
%C Product over all primes of highest power of prime less than or equal to n. a(0) = 1 by convention.
%C Also smallest number whose set of divisors contains an n-term arithmetic progression. - _Reinhard Zumkeller_, Dec 09 2002
%C 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)
%C Corollary 3 of Farhi gives a proof that a(n) >= 2^(n-1). - _Jonathan Vos Post_, Jun 15 2009
%C Appears to be row products of the triangle T(n,k) = b(A010766) where b = A130087/A130086. - _Mats Granvik_, Jul 08 2009
%C 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
%C a(n) = lcm(A188666(n), A188666(n)+1, ..., n). - _Reinhard Zumkeller_, Apr 25 2011
%C 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
%C It appears that A020500(n) = a(n)/a(n-1). - _Asher Auel_, corrected by _Bill McEachen_, Apr 05 2024
%C n-th distinct value = A051451(n). - _Matthew Vandermast_, Nov 27 2009
%C a(n+1) = least common multiple of n-th row in A213999. - _Reinhard Zumkeller_, Jul 03 2012
%C For n > 2, (n-1) = Sum_{k=2..n} exp(a(n)*2*i*Pi/k). - _Eric Desbiaux_, Sep 13 2012
%C First column minus second column of A027446. - _Eric Desbiaux_, Mar 29 2013
%C 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
%C Slowest growing integer > 0 in Z converging to 0 in Z^ when considered as profinite integer. - _Herbert Eberle_, May 01 2016
%C 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
%C 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
%C 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
%C 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
%C 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
%D J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 365.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A003418/b003418.txt">Table of n, a(n) for n = 0..2308</a> (first 501 terms from T. D. Noe)
%H R. Anderson and N. J. A. Sloane, <a href="/A003418/a003418_1.pdf">Correspondence, 1975</a>.
%H Dorin Andrica, Sorin Rădulescu, and George Cătălin Ţurcaş, <a href="https://doi.org/10.1007/978-3-030-55857-4_4">The Exponent of a Group: Properties, Computations and Applications</a>, Disc. Math. and Applications, Springer, Cham (2020), 57-108.
%H Javier Cilleruelo, Juanjo Rué, Paulius Šarka, and Ana Zumalacárregui, <a href="http://arxiv.org/abs/1112.3013">The least common multiple of sets of positive integers</a>, arXiv:1112.3013 [math.NT], 2011.
%H R. E. Crandall and C. Pomerance, Prime numbers: a computational perspective, <a href="http://www.ams.org/mathscinet-getitem?mr=2156291">MR2156291</a>, p. 61.
%H Roger B. Eggleton, <a href="https://www.jstor.org/stable/2690332">Least Common Multiple of {1,2,...,n}</a>, Mathematics Magazine, 61(1) (1988), pp. 47-48, Problem 1252.
%H Bakir Farhi, <a href="http://arxiv.org/abs/0906.2295">An identity involving the least common multiple of binomial coefficients and its application</a>, arXiv:0906.2295 [math.NT], 2009.
%H Bakir Farhi, <a href="https://www.jstor.org/stable/40391302">An identity involving the least common multiple of binomial coefficients and its application</a>, Amer. Math. Monthly 116(9) (2009), 836-839.
%H Steven Finch, <a href="/A003418/a003418.pdf">Cilleruelo's LCM Constants</a>, 2013. [Cached copy, with permission of the author]
%H V. L. Gavrikov, <a href="https://arxiv.org/abs/1806.09264">On property of least common multiple to be a D-magic number</a>, arXiv:1806.09264 [math.NT], 2018.
%H S. Labbé and E. Pelantová, <a href="http://arxiv.org/abs/1409.7510">Palindromic sequences generated from marked morphisms</a>, arXiv:1409.7510 [math.CO], 2014.
%H J. C. Lagarias, <a href="https://www.jstor.org/stable/2695443">An elementary problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 109 (6) (2002) 534-543. <a href="https://arxiv.org/abs/math/0008177">arXiv:math/0008177 [math.NT]</a>, 2000-2001.
%H P. Luschny and S. Wehmeier, <a href="http://arxiv.org/abs/0909.1838">The lcm(1, 2, ..., n) as a product of sine values sampled over the points in Farey sequences</a>, arXiv:0909.1838 [math.CA], 2009.
%H Des MacHale and Joseph Manning, <a href="http://dx.doi.org/10.1017/mag.2015.28">Maximal runs of strictly composite integers</a>, The Mathematical Gazette, 99, pp 213-219 (2015).
%H Greg Martin, <a href="http://arxiv.org/abs/0907.4384">A product of Gamma function values at fractions with the same denominator</a>, arXiv:0907.4384 [math.CA], 2009.
%H M. Nair, <a href="https://www.jstor.org/stable/2320934">On Chebychev-type inequalities for primes</a> Amer. Math. Monthly 89(2) (1982), 126-129.
%H S. Ramanujan, <a href="https://doi.org/10.1112/plms/s2_14.1.347">Highly composite numbers</a>, 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 <a href="http://ramanujan.sirinudi.org/Volumes/published/ram15.html">here</a>.)
%H E. S. Selmer, <a href="http://www.mscand.dk/article/view/11662/9678">On the number of prime divisors of a binomial coefficient</a>, Math. Scand. 39 (1976), no. 2, 271-281 (1977).
%H Jonathan Sondow, <a href="http://dx.doi.org/10.1090/S0002-9939-03-07081-3">Criteria for irrationality of Euler's constant</a>, Proc. AMS 131 (2003), 3335.
%H Rosemary Sullivan and Neil Watling, <a href="http://www.emis.de/journals/INTEGERS/papers/n65/n65.Abstract.html">Independent divisibility pairs on the set of integers from 1 to n</a>, INTEGERS 13 (2013) #A65.
%H M. Tchebichef, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k163969/f374">Mémoire sur les nombres premiers</a>, J. Math. Pures Appliquées 17 (1852), 366-390.
%H Helge von Koch, <a href="http://dx.doi.org/10.1007/BF02403071">Sur la distribution des nombres premiers</a>, Acta Math. 24 (1) (1901), 159-182.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LeastCommonMultiple.html">Least Common Multiple</a>, <a href="http://mathworld.wolfram.com/ChebyshevFunctions.html">Chebyshev Functions</a>, <a href="http://mathworld.wolfram.com/MangoldtFunction.html">Mangoldt Function</a>.
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>
%F 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
%F 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
%F 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
%F Recursive formula useful for computations: a(0)=1; a(1)=1; a(n)=lcm(n,a(n-1)). - _Enrique Pérez Herrero_, Jan 08 2011
%F From _Enrique Pérez Herrero_, Jun 01 2011: (Start)
%F a(n)/a(n-1) = A014963(n).
%F if n is a prime power p^k then a(n)=a(p^k)=p*a(n-1), otherwise a(n)=a(n-1).
%F a(n) = Product_{k=2..n} (1 + (A007947(k)-1)*floor(1/A001221(k))), for n > 1. (End)
%F a(n) = A079542(n+1, 2) for n > 1.
%F a(n) = exp(Sum_{k=1..n} Sum_{d|k} moebius(d)*log(k/d)). - _Peter Luschny_, Sep 01 2012
%F a(n) = A025529(n) - A027457(n). - _Eric Desbiaux_, Mar 14 2013
%F 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
%F a(n) = A064446(n)*A038610(n). - _Anthony Browne_, Jun 16 2016
%F a(n) = A000142(n) / A025527(n) = A000793(n) * A225558(n). - _Antti Karttunen_, Jun 02 2017
%F log(a(n)) = Sum_{k>=1} (A309229(n, k)/k - 1/k). - _Mats Granvik_, Aug 10 2019
%F From _Petros Hadjicostas_, Jul 24 2020: (Start)
%F Nair (1982) proved that 2^n <= a(n) <= 4^n for n >= 9. See also Farhi (2009). Nair also proved that
%F a(n) = lcm(m*binomial(n,m): 1 <= m <= n) and
%F a(n) = gcd(a(m)*binomial(n,m): n/2 <= m <= n). (End)
%F Sum_{n>=1} 1/a(n) = A064859. - _Bernard Schott_, Aug 24 2020
%e 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.
%e floor(log(6)/log(3)) = 1 so the exponent of 3 is 1.
%e 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
%p A003418 := n-> lcm(seq(i,i=1..n));
%p 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_
%p # next Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1, ilcm(n, a(n-1))) end:
%p seq(a(n), n=0..33); # _Alois P. Heinz_, Jun 10 2021
%t Table[LCM @@ Range[n], {n, 1, 40}] (* _Stefan Steinerberger_, Apr 01 2006 *)
%t FoldList[ LCM, 1, Range@ 28]
%t A003418[0] := 1; A003418[1] := 1; A003418[n_] := A003418[n] = LCM[n,A003418[n-1]]; (* _Enrique Pérez Herrero_, Jan 08 2011 *)
%t Table[Product[Prime[i]^Floor[Log[Prime[i], n]], {i, PrimePi[n]}], {n, 0, 28}] (* _Wei Zhou_, Jun 25 2011 *)
%t Table[Product[Cyclotomic[n, 1], {n, 2, m}], {m, 0, 28}] (* _Fred Daniel Kline_, May 22 2014 *)
%t a1[n_] := 1/12 (Pi^2+3(-1)^n (PolyGamma[1,1+n/2] - PolyGamma[1,(1+n)/2])) // Simplify
%t a[n_] := Denominator[Sqrt[a1[n]]];
%t Table[If[IntegerQ[a[n]], a[n], a[n]*(a[n])[[2]]], {n, 0, 28}] (* _Gerry Martens_, Apr 07 2018 [Corrected by _Vaclav Kotesovec_, Jul 16 2021] *)
%o (PARI) a(n)=local(t); t=n>=0; forprime(p=2,n,t*=p^(log(n)\log(p))); t
%o (PARI) a(n)=if(n<1,n==0,1/content(vector(n,k,1/k)))
%o (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
%o (PARI) a(n)=lcm(vector(n,i,i)) \\ Bill Allombert, Apr 18 2012 [via _Charles R Greathouse IV_]
%o (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
%o (Sage) [lcm(range(1,n)) for n in range(1, 30)] # _Zerinvary Lajos_, Jun 06 2009
%o (Haskell)
%o a003418 = foldl lcm 1 . enumFromTo 2
%o -- _Reinhard Zumkeller_, Apr 04 2012, Apr 25 2011
%o (Magma) [1] cat [Exponent(SymmetricGroup(n)) : n in [1..28]]; // _Arkadiusz Wesolowski_, Sep 10 2013
%o (Magma) [Lcm([1..n]): n in [0..30]]; // _Bruno Berselli_, Feb 06 2015
%o (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
%o (Python)
%o from functools import reduce
%o from operator import mul
%o from sympy import sieve
%o def integerlog(n,b): # find largest integer k>=0 such that b^k <= n
%o kmin, kmax = 0,1
%o while b**kmax <= n:
%o kmax *= 2
%o while True:
%o kmid = (kmax+kmin)//2
%o if b**kmid > n:
%o kmax = kmid
%o else:
%o kmin = kmid
%o if kmax-kmin <= 1:
%o break
%o return kmin
%o def A003418(n):
%o return reduce(mul,(p**integerlog(n,p) for p in sieve.primerange(1,n+1)),1) # _Chai Wah Wu_, Mar 13 2021
%o (Python) # generates initial segment of sequence
%o from math import gcd
%o from itertools import accumulate
%o def lcm(a, b): return a * b // gcd(a, b)
%o def aupton(nn): return [1] + list(accumulate(range(1, nn+1), lcm))
%o print(aupton(30)) # _Michael S. Branicky_, Jun 10 2021
%Y Row products of A133233.
%Y 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.
%Y Cf. A025528 (number of prime factors of a(n) with multiplicity).
%Y Cf. A275120 (lengths of runs of consecutive equal terms), A276781 (ordinal transform from term a(1)=1 onward).
%K nonn,easy,core,nice
%O 0,3
%A Roland Anderson (roland.anderson(AT)swipnet.se)
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