A205959 a(n) = n^omega(n)/rad(n).

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

Offset: 1

Author

Peter Luschny, Feb 03 2012

Keywords

• nonn
• nice

Comments

a(n) = exp(-Sum_{d in P} moebius(d)*log(n/d)) where P = {d : d divides n and d is prime}. This is a variant of the (exponential of the) von Mangoldt function where the divisors are restricted to prime divisors. The (exponential of the) summatory function is A205957. Apart from n=1 the value is 1 if and only if n is prime; the fixed points are the products of two distinct primes (A006881).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Peter Luschny, The von Mangoldt Transformation.

Crossrefs

Cf. A003418, A025527, A008578, A102467, A006881, A205957.

Programs

Haskell

a205959 n = product $ map (div n) $ a027748_row n
-- Reinhard Zumkeller, Dec 15 2013

Maple

with(numtheory): A205959 := proc(n) select(isprime, divisors(n));
simplify(exp(-add(mobius(d)*log(n/d), d=%))) end:
# Alternative:
a := n -> local p; mul(n/p[1], p in ifactors(n)[2]):
seq(a(n), n = 1..68); # Peter Luschny, Jul 19 2023

Mathematica

a[n_] := Exp[-Sum[ MoebiusMu[d]*Log[n/d], {d, FactorInteger[n][[All, 1]]}]]; Table[a[n], {n, 1, 68}] (* Jean-François Alcover, Jan 15 2013 *)

PARI

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

Python

from math import prod
from sympy import primefactors
def A205959(n): return prod(n//p for p in primefactors(n)) # Chai Wah Wu, Jul 12 2023

Sage

def A205959(n) :
    P = filter(is_prime, divisors(n))
    return simplify(exp(-add(moebius(d)*log(n/d) for d in P)))
[A205959(n) for n in (1..60)]

Formula

a(n) = Product_{p|n} n/p. - Charles R Greathouse IV, Jun 27 2013

a(n) = Product_{k=1..A001221(n)} n/A027748(n,k). - Reinhard Zumkeller, Dec 15 2013

If n is squarefree, then a(n) = n^(omega(n)-1). - Wesley Ivan Hurt, Jun 09 2020

a(p^e) = p^(e-1) for p prime, e > 0. - Bernard Schott, Jun 09 2020

Extensions

New name from Charles R Greathouse IV, Jun 30 2013