Recently Greg Martin (arXiv:0907.4384) showed a cute little formula
for the lcm{1,2,3,..,n} = LCM(n).

Jonathan Vos Post reported this on OEIS in A003418 as follows:

*"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))*LCM(n)^(-1/2)."
*Thus we look at a Farey sequence. If a/b is a member of this sequence 1−a/b
is also a member. We have to compute a product and so we can trade in two Gamma
evaluations for one sine evaluation by the reflection formula.

Therefore let's look at two more items: the number of terms in a Farey sequence of order n, which is Sum{i=1..n} φ(i) − 1 (see A015614). And a nice cup of tea.

Here φ(n) denotes Euler's totient function, which is the number of positive integers not exceeding n and relatively prime to n. Further let Λ(n) denote von Mangoldt's function, then the second equation can be rewritten as

The last term, Sum lbs(k), can be approximated by

prox(x) = x( 3x / Pi^2 − 1) − sqrt(x) log(x).

The plot below shows the difference between the true value and the approximation for 3 ≤ n ≤ 3000.

See the Wikipedia article for more on Farey sequences.

OEIS | seq | |

Number of fractions in a Farey sequence of order n |
A005728 |
1, 2, 3, 5, 7, 11, 13, 19 |

Sum of phi | A002088 |
0, 1, 2, 4, 6, 10, 12, 18 |

Lcm of {1,2,3,...n} | A003418 |
1, 1, 2, 6, 12, 60, 60 |

Lcm of the proper divisors of n | A048671 |
1,1,1, 2,1, 6, 1, 4,3, 10 |

Exponential von Mangoldt | A014963 |
1, 2, 3, 2, 5, 1, 7, 2, 3 |

Not a prime power | A024619 |
6, 10, 12, 14, 15, 18, 20 |

2^phi | A066781 |
2, 2, 4, 4, 16, 4, 64, 16 |

Prime powers | A000961 |
1, 2, 3, 4, 5, 7, 8, 9 |

Twice the prime powers | A138929 |
2, 4, 6, 8, 10, 14, 16,18 |

Cyclotomic polynomials at x = 1 | A020500 |
0, 2, 3, 2, 5, 1, 7, 2, 3 |

Cyclotomic polynomials at x =-1 | A020513 |
-1, -2,0,1,2,1,3,1,2,1, 5 |

Here is a Maple worksheet.