A002944 -id:A002944 - cp's OEIS frontend

A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.

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

Offset: 0

Roland Anderson (roland.anderson(AT)swipnet.se)

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)
Corollary 3 of Farhi gives a proof that a(n) >= 2^(n-1). - 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)/a(n-1). - Asher Auel, corrected by Bill McEachen, Apr 05 2024
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(a(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

			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

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).

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.
Peter 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.
Index to divisibility sequences
Index entries for "core" sequences
Index entries for sequences related to lcm's

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).

a003418 = foldl lcm 1 . enumFromTo 2
-- Reinhard Zumkeller, Apr 04 2012, Apr 25 2011

[1] cat [Exponent(SymmetricGroup(n)) : n in [1..28]]; // Arkadiusz Wesolowski, Sep 10 2013

[Lcm([1..n]): n in [0..30]]; // Bruno Berselli, Feb 06 2015

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

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 *)
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])[[2]]], {n, 0, 28}] (* Gerry Martens, Apr 07 2018 [Corrected by Vaclav Kotesovec, Jul 16 2021] *)

a(n)=local(t); t=n>=0; forprime(p=2,n,t*=p^(log(n)\log(p))); t

a(n)=if(n<1,n==0,1/content(vector(n,k,1/k)))

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

a(n)=lcm(vector(n,i,i)) \\ Bill Allombert, Apr 18 2012 [via Charles R Greathouse IV]

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

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

# 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 [1] + list(accumulate(range(1, nn+1), lcm))
print(aupton(30)) # Michael S. Branicky, Jun 10 2021

[lcm(range(1,n)) for n in range(1, 30)] # Zerinvary Lajos, Jun 06 2009

(define (A003418 n) (let loop ((n n) (m 1)) (if (zero? n) m (loop (- n 1) (lcm m n))))) ;; Antti Karttunen, Jan 03 2018

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

A276781 a(n) = 1+n-(nearest power of prime <= n); for n > 1, a(n) = minimal b such that the numbers binomial(n,k) for b <= k <= n-b have a common divisor greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 3, 1, 2, 3, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 1, 2, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 1, 2

Offset: 1

N. J. A. Sloane, Sep 29 2016, following a suggestion from Eric Desbiaux

The definition in the video has "b < k < n-b" rather than "b <= k <= n-b", but that appears to be a typographical error.
From Antti Karttunen, Jan 21 2020: (Start)
a(n) = 1 if n is a power of prime (term of A000961), otherwise a(n) is one more than the distance to the nearest preceding prime power.
For n > 1, a(n) indicates the maximum region on the row n of Pascal's triangle (A007318) such that binomial terms C(n,a(n)) .. C(n,n-a(n)) all share a common prime factor. Because for all prime powers, p^k, the binomial terms C(p^k,1) .. C(p^k,p^k-1) have p as their prime factor, we have a(A000961(n)) = 1 for all n, while for each successive n that is not a prime power, the region of shared prime factor shrinks one step more towards the center of the triangle. From this follows that this is the ordinal transform of A025528 (equally, of A065515, or of A003418(n) from n >= 1 onward), equivalent to the simple definition given above.
(End)

			Row 6 of Pascal's triangle is 1,6,15,20,15,6,1 and [15,20,15] have a common divisor of 5. Since 15 = binomial(6,2), a(6)=2.

Antti Karttunen, Table of n, a(n) for n = 1..16384 (terms for n = 2..1685 from R. J. Mathar, terms 1686..10000 from Chai Wah Wu).
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Christophe Soulé, Le triangle de Pascal et ses propriétés, Lecture, Soc. Math. de France, Feb 13 2008; SMF link (in French).

Cf. A007318, A010055, A276782 (positions of records), A000961 (positions of ones), A024619 (positions of terms > 1).

Cf. A002944, A003418, A025528, A065515, A175851.

mygcd:=proc(lis) local i,g,m;
m:=nops(lis); g:=lis[1];
for i from 2 to m do g:=gcd(g,lis[i]); od:
g; end;
f:=proc(n) local b,lis; global mygcd;
for b from floor(n/2) by -1 to 1 do
lis:=[seq(binomial(n,i),i=b..n-b)];
if mygcd(lis)=1 then break; fi; od:
b+1;
end;
[seq(f(n),n=2..120)];

Table[b = 1; While[GCD @@ Map[Binomial[n, #] &, Range[b, n - b]] == 1, b++]; b, {n, 92}] (* Michael De Vlieger, Oct 03 2016 *)

A276781(n) = if(1==n,1,forstep(k=n,1,-1,if(isprimepower(k),return(1+n-k)))); \\ Antti Karttunen, Jan 21 2020

from sympy import factorint
def A276781(n): return 1+n-next(filter(lambda m:len(factorint(m))<=1, range(n,0,-1))) # Chai Wah Wu, Oct 25 2024

If A010055(n) == 1, a(n) = 1, otherwise a(n) = 1 + a(n-1). - Antti Karttunen, Jan 21 2020

Term a(1) = 1 prepended and alternative simpler definition added to the name by Antti Karttunen, Jan 20 2020

A025527 a(n) = n!/lcm{1,2,...,n} = (n-1)!/lcm{C(n-1,0), C(n-1,1), ..., C(n-1,n-1)}.

Original entry on oeis.org

1, 1, 1, 2, 2, 12, 12, 48, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 522547200, 522547200, 10450944000, 219469824000, 4828336128000, 4828336128000, 115880067072000, 579400335360000, 15064408719360000

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

a(n) = a(n-1) iff n is prime. Thus a(1)=a(2)=a(3)=1 is the only triple in this sequence. - Franz Vrabec, Sep 10 2005
a(k) = a(k+1) for k in A006093. - Lekraj Beedassy, Aug 03 2006
Partial products of A048671. - Peter Luschny, Sep 09 2009

			a(5) = 2 as 5!/lcm(1..5) = 120/60 = 2.

Alois P. Heinz, Table of n, a(n) for n = 1..500
Liam Solus, Simplices for Numeral Systems, arXiv:1706.00480 [math.CO], 2017. Mentions this sequence.
Index entries for sequences related to lcm's

Cf. A000142, A002541, A006093, A003418, A048671, A025529, A205957.

A048671 a(n) is the least common multiple of the proper divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 4, 3, 10, 1, 12, 1, 14, 15, 8, 1, 18, 1, 20, 21, 22, 1, 24, 5, 26, 9, 28, 1, 30, 1, 16, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 7, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63, 32, 65, 66, 1, 68, 69, 70, 1, 72, 1, 74, 75, 76, 77, 78, 1

Offset: 1

Labos Elemer

A proper divisor d of n is a divisor of n such that 1 <= d < n.

Previous name was: a(n) = q(n)/q(n-1), where q(n) = n!/A003418(n).

			8!/lcm(8) = 48 = 40320/840 while 7!/lcm(7) = 5040/420 = 12 so a(8) = 48/12 = 4.
a(5) = 1 = lcm(1,2,3,4,5)/lcm(1,5,10,10,5,1).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Peter Luschny and Stefan 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.
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.
Index entries for sequences related to lcm's

Cf. A000142, A002944, A003418, A014963, A025527.

Cf. A182936 gives the dual (greatest common divisor).

A048671 := n -> ilcm(op(numtheory[divisors](n) minus {1,n}));
seq(A048671(i), i=1..79); # Peter Luschny, Mar 21 2011

{1}~Join~Table[LCM @@ Most@ Divisors@ n, {n, 2, 79}] (* Michael De Vlieger, May 01 2016 *)

a(n)=my(p=n);if(isprime(n)||(ispower(n,,&p)&&isprime(p)),n/p,n) \\ Charles R Greathouse IV, Jun 24 2011

a(n)=my(p); if(isprimepower(n,&p), n/p, n) \\ Charles R Greathouse IV, May 02 2016

def A048671(n) :
    if n < 2 : return 1
    else : D = divisors(n); D.pop()
    return lcm(D)
[A048671(i) for i in (1..79)] # Peter Luschny, Feb 03 2012

a(n) = A025527(n)/A025527(n-1).
a(n) = (n*A003418(n-1))/A003418(n).
a(n) = A003418(n-1)/A002944(n). [corrected by Michel Marcus, May 18 2020]
From Henry Bottomley, May 19 2000: (Start)
a(n) = n/A014963(n) = lcm(A052126(n), A032742(n)).
a(n) = n if n not a prime power, a(n) = n/p if n = p^m (i.e., a(n) = 1 if n = p). (End)
From Vladeta Jovovic, Jul 04 2002: (Start)
a(n) = n*Product_{d | n} d^mu(d).
Product_{d | n} a(d) = A007956(n). (End)
a(n) = Product_{k=1..n-1} if(gcd(n, k) > 1, 1 - exp(2*pi*i*k/n), 1), where i = sqrt(-1). - Paul Barry, Apr 15 2005
From Peter Luschny, Jun 09 2011: (Start)
a(n) = Product_{k=1..n-1} if(gcd(k,n) > 1, 2*Pi/Gamma(k/n)^2, 1).
a(n) = Product_{k=1..n-1} if(gcd(k,n) > 1, 2*sin(Pi*k/n), 1). (End)

New definition based on a comment of David Wasserman by Peter Luschny, Mar 23 2011

cp's OEIS Frontend

A025537 a(n) = (1/s(1) + 1/s(2) + ... + 1/s(n+1)) * LCM{1, 2, ..., n}, where s(k) = LCM{1,2,...,k}/k = A002944(k).

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A081529 Duplicate of A002944.

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A092593 a(n) is the smallest number k > 1 for which A001142(k)/A002944(k+1)^n is an integer.

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A025538 a(n) = (1/s(1) - 1/s(2) + ... + d/s(n+1)) * LCM{1, 2, ..., n}, where d = (-1)^n, s = A002944, i.e., s(k) = LCM of row k of Pascal's triangle.

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A056611 Quotient: squarefree kernel of A002944(n) divided by that of A001405.

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A092592 a(n) = A001142(n)/A002944(n), i.e., the product of C(n,j) binomial coefficients (for j=0..n) is divided by the least common multiple of them.

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A003418 Least common multiple (or LCM) of {1, 2, ..., n} for n >= 1, a(0) = 1.

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A276781 a(n) = 1+n-(nearest power of prime <= n); for n > 1, a(n) = minimal b such that the numbers binomial(n,k) for b <= k <= n-b have a common divisor greater than 1.

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