A205957 a(n) = exp(-Sum_{k=1..n} Sum_{d|k, d prime} moebius(d)*log(k/d)).

1, 1, 1, 1, 2, 2, 12, 12, 48, 144, 1440, 1440, 34560, 34560, 483840, 7257600, 58060800, 58060800, 3135283200, 3135283200, 125411328000, 2633637888000, 57940033536000, 57940033536000, 5562243219456000, 27811216097280000, 723091618529280000, 6507824566763520000

Offset: 0

Author

Peter Luschny, Sep 01 2012

Keywords

• nonn

Comments

The author proposes to denote this sequence lcm_{p}(n) as lcm(n) = lcm({1,2,..n}) = exp(Sum_{k=1..n} Sum_{d|k} moebius(d)*log(k/d)).

For n > 0 the a(n) are the partial products of A205959(n), which is the exponential of a modified von Mangoldt function where the divisors are restricted to prime divisors.

Links

• Michel Marcus, Table of n, a(n) for n = 0..400
• Peter Luschny, The von Mangoldt Transformation.

Crossrefs

Cf. A003418, A205959, A216152, A216153.

Programs

Maple

with(numtheory):
A205957 := proc(n) simplify(exp(-add(add(mobius(d)*log(k/d), d=select(isprime, divisors(k))),k=1..n))) end: seq(A205957(i), i=0..27);

Mathematica

a[n_] := Exp[-Sum[ MoebiusMu[p] Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]]; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Jun 27 2013 *)

PARI

a(n)=prod(k=4,n,my(f=factor(k)[, 1]); prod(i=1, #f, k/f[i])) \\ Charles R Greathouse IV, Jun 27 2013

Sage

def A205957(n) : return simplify(exp(-add(add(moebius(p)*log(k/p) for p in prime_divisors(k)) for k in (1..n))))

Formula

a(n) = Product_{p prime, p<=n} (floor(n/p)!). - Ridouane Oudra, Nov 22 2021