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A205959 a(n) = n^omega(n)/rad(n). 5


%S 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,

%T 900,1,16,33,34,35,216,1,38,39,160,1,1764,1,88,135,46,1,384,7,250,51,

%U 104,1,486,55,224,57,58,1,7200,1,62,189,32,65,4356,1,136

%N a(n) = n^omega(n)/rad(n).

%C 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).

%H Reinhard Zumkeller, <a href="/A205959/b205959.txt">Table of n, a(n) for n = 1..10000</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/VonMangoldtTransformation">The von Mangoldt Transformation</a>.

%F a(n) = prod_{p|n} n/p. - _Charles R Greathouse IV_, Jun 27 2013

%F a(n) = prod(n/A027748(n,k): k=1..A001221(n)). - _Reinhard Zumkeller_, Dec 15 2013

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

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

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

%t a[n_] := Exp[-Sum[ MoebiusMu[d]*Log[n/d], {d, FactorInteger[n][[All, 1]]}]]; Table[a[n], {n, 1, 68}] (* _Jean-Fran├žois Alcover_, Jan 15 2013 *)

%o (Sage)

%o def A205959(n) :

%o P = filter(is_prime, divisors(n))

%o return simplify(exp(-add(moebius(d)*log(n/d) for d in P)))

%o [A205959(n) for n in (1..60)]

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

%o (Haskell)

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

%o -- _Reinhard Zumkeller_, Dec 15 2013

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

%K nonn,nice

%O 1,4

%A _Peter Luschny_, Feb 03 2012

%E New name from _Charles R Greathouse IV_, Jun 30 2013

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Last modified November 22 16:22 EST 2019. Contains 329396 sequences. (Running on oeis4.)