A216152 -id:A216152 - cp's OEIS frontend

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

Peter Luschny, Sep 01 2012

nonn

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.

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

Cf. A003418, A205959, A216152, A216153.

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);

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 *)

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

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

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

A216153 The partial products of a(n) are the distinct values of the exponential of the von Mangoldt function modified by restricting the divisors to prime divisors (A205957).

Original entry on oeis.org

1, 2, 6, 4, 3, 10, 24, 14, 15, 8, 54, 40, 21, 22, 96, 5, 26, 9, 56, 900, 16, 33, 34, 35, 216, 38, 39, 160, 1764, 88, 135, 46, 384, 7, 250, 51, 104, 486, 55, 224, 57, 58, 7200, 62, 189, 32, 65, 4356, 136, 69, 4900, 864, 74, 375, 152, 77, 6084, 640, 27, 82

Offset: 1

Peter Luschny, Sep 02 2012

The partial products of a(n) are A216152(n) which are the distinct values of the 'prime lcm(n)' A205957.
Let b(n) denote the nonprime numbers A018252(n).
If n = 1 then a(n) = b(n) = 1
else if a(n) < b(n) then
a(n) is a cototient of consecutive pure powers of primes (A053211),
b(n) is a prime power with exponent > 1 (A025475),
b(n)/a(n) is a prime root of n-th nontrivial prime power (A025476);
else if a(n) > b(n) then
b(n) is a number which is neither a prime power nor a semiprime (A102467);
else if a(n) = b(n) then
a(n) is the product of two distinct primes (A006881).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Peter Luschny, The von Mangoldt Transformation.

Cf. A205957, A205959, A216152.

A205957[n_] := Exp[-Sum[ MoebiusMu[p]*Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]]; nonPrime[1] = 1; nonPrime[n_] := Which[k0 = k /. FindRoot[ n + PrimePi[k] == k , {k, n}] // Floor; n+PrimePi[k0] == k0, k0 , n+PrimePi[k0+1] == k0+1, k0+1, n+PrimePi[k0+2] == k0+2, k0+2, True, k0]; a[1] = 1; a[n_] := A205957[nonPrime[n]] / A205957[nonPrime[n-1]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jun 27 2013 *)

def A216153(n):
    if n == 1 : return 1
    return A205957(A018252(n))/A205957(A018252(n-1))

a(n) = A205957(A018252(n))/A205957(A018252(n-1)) for n > 1, a(1) = 1.

cp's OEIS Frontend

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

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