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A205959 a(n) = n^omega(n)/rad(n). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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.

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

MAPLE

with(numtheory): A205959 := proc(n) select(isprime, divisors(n));

simplify(exp(-add(mobius(d)*log(n/d), d=%))) end:

seq(A205959(i), i=1..60);

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 *)

PROG

(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)]

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

(Haskell)

a205959 n = product $ map (div n) $ a027748_row n

-- Reinhard Zumkeller, Dec 15 2013

CROSSREFS

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

Sequence in context: A349440 A048671 A335023 * A318503 A328580 A088123

Adjacent sequences: A205956 A205957 A205958 * A205960 A205961 A205962

KEYWORD

nonn,nice

AUTHOR

Peter Luschny, Feb 03 2012

EXTENSIONS

New name from Charles R Greathouse IV, Jun 30 2013

STATUS

approved

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Last modified December 8 07:37 EST 2022. Contains 358691 sequences. (Running on oeis4.)