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%I #46 Jul 19 2023 05:35:16
%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) = Product_{p|n} n/p. - _Charles R Greathouse IV_, Jun 27 2013
%F a(n) = Product_{k=1..A001221(n)} n/A027748(n,k). - _Reinhard Zumkeller_, Dec 15 2013
%F If n is squarefree, then a(n) = n^(omega(n)-1). - _Wesley Ivan Hurt_, Jun 09 2020
%F a(p^e) = p^(e-1) for p prime, e > 0. - _Bernard Schott_, Jun 09 2020
%p with(numtheory): A205959 := proc(n) select(isprime, divisors(n));
%p simplify(exp(-add(mobius(d)*log(n/d), d=%))) end:
%p # Alternative:
%p a := n -> local p; mul(n/p[1], p in ifactors(n)[2]):
%p seq(a(n), n = 1..68); # _Peter Luschny_, Jul 19 2023
%t 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 *)
%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
%o (Python)
%o from math import prod
%o from sympy import primefactors
%o def A205959(n): return prod(n//p for p in primefactors(n)) # _Chai Wah Wu_, Jul 12 2023
%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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