A093880 a(n) = lcm(1, 2, ..., 2n) / lcm(1, 2, ..., n).

1, 2, 6, 10, 70, 42, 462, 858, 858, 4862, 92378, 8398, 193154, 74290, 222870, 6463230, 200360130, 11785890, 11785890, 22951470, 22951470, 941010270, 40463441610, 1759280070, 82686163290, 115760628606, 115760628606, 2045104438706

Offset

0,2

Comments

Also, a(n) = lcm(n+1, n+2, ..., 2n-1, 2n) / lcm(1, 2, ..., n-1, n).

Links

T. D. Noe, Table of n, a(n) for n = 0..500

Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003) 3335-3344.

Eric Weisstein's World of Mathematics, Least Common Multiple

Index entries for sequences related to lcm's

Formula

a(n) = A003418(2n)/A003418(n) = A099996(n)/A003418(n).

The prime number theorem implies that a(n) = e^(n(1+o(1))) as n -> infinity. In other words, log(a(n))/n -> 1 as n -> infinity. - Jonathan Sondow, Jan 17 2005

a(n) = a(n-1)*A014963(2*n-1)/A155457(n) for n >= 2. - Peter Luschny, Jan 01 2026

Example

The LCM of {1,2,3,4,5,6} is 60 and the LCM of {1,2,3} is 6, so a(3) = 60/6 = 10.

Maple

a := n -> lcm(seq(j, j=n+1..2*n)) / lcm(seq(j, j=1..n)):
seq(a(n), n=0..32); # Emeric Deutsch, Feb 02 2006

Mathematica

a[0] := 1;  a[n_] := LCM @@ Range[n + 1, 2 n] / LCM @@ Range[n];
Table[a[n], {n, 0, 27}] (* Robert G. Wilson v, Jan 22 2005 *)

Prog

(Python)
from math import lcm
def a(n): return lcm(*range(n+1, 2*n+1)) // lcm(*range(1, n+1)) # David Radcliffe, Dec 30 2025
(SageMath)
def power_base(n: int) -> int:
    b, e = is_prime_power(n, get_data=True)
    return b if e != 0 else 1
@cached_function
def a(n: int) -> int:
    if n < 2: return n + 1
    b2 = power_base(2 * n - 1)
    b1 = power_base(n)
    ret = a(n-1) * b2 if b2 > 1 else a(n-1)
    return ret // b1 if b1 > 1 and b1 % 2 != 0 else ret
print([a(n) for n in range(28)])  # Peter Luschny, Jan 01 2026

Crossrefs

Cf. A003418, A099996, A080397, A014963, A155457.

Sequence in context: A163788 A361792 A324547 * A080397 A322756 A048782

Adjacent sequences: A093877 A093878 A093879 * A093881 A093882 A093883

Keyword

nonn

Author

Amarnath Murthy, Apr 22 2004

Extensions

More terms from Emeric Deutsch, Feb 02 2006

Entry revised by N. J. A. Sloane, Jan 24 2007

a(0)=1 prepended by David Radcliffe, Dec 30 2025

Status

approved