%I #36 Jul 19 2023 01:02:19

%S 1,1,1,4,1,1,1,64,9,1,1,12,1,1,1,4096,1,18,1,20,1,1,1,576,25,1,729,28,

%T 1,1,1,1048576,1,1,1,1296,1,1,1,1600,1,1,1,44,45,1,1,110592,49,50,1,

%U 52,1,2916,1,3136,1,1,1,60,1,1,63,1073741824,1,1,1,68

%N a(n) = n^excess(n), where excess(n) = A046660(n).

%H Plot2 <a href="/A363919/a363919.jpg">A363923 vs A205959</a>.

%H Michael De Vlieger, <a href="/A363919/a363919.png">Plot p(k)^e(k) at (x,y) = (n,k)</a> for n = 1..2^11, with a color function representing e(k) where black represents e(k) = 1, red e(k) = 2, through magenta. The bar at bottom represents a(n) = 1 in black, a(n) a composite prime power in gold, a(n) neither squarefree nor semiprime in blue, and a(n) such that all e(k) > 1 in purple.

%H Michael De Vlieger, <a href="/A363919/a363919.pdf">Notes on certain sequences relating BigOmega(n), omega(n), and rad(n)</a>, 2023.

%F a(n) = n^(Sum_{p in Factors(n)} (mult(p) - 1)), where Factors(n) is the integer factorization of n and mult(p) the multiplicity of the prime factor p.

%F a(n) = A363923(n) / A205959(n).

%F a(n) = n^A046660(n) = n^(A001222(n) - A001221(n)).

%F a(n) = 1 or divisible by at least one squared prime.

%F a(n) = 1 <=> n is squarefree (A005117).

%F a(n) != 1 <=> A056170(n) != 0.

%F a(n) = n <=> n = A060687(n - 1) for n >= 2.

%F a(2^n) = 2^(n*(n - 1)) = A053763(n).

%F a(n) <= 2^(lb(n)*(lb(n)-1)), where lb(n) = floor(log_{2}(n)).

%F a(n) is even <=> n = 2*A337945(n).

%F a(n) > 1 is odd <=> n = A053850(n).

%F n is prime => a(n) = 1. ('prime' means term of A000040).

%F n is prime product => a(n) = 1. ('prime product' means term of A144338).

%F n is proper prime power => a(n) is proper prime power. ('proper prime power' means term of A246547).

%F Moebius(a(n)) = [a(n) = 1], where [ ] denotes the Iverson bracket.

%e 108 = 2^2 * 3^3 => excess(108) = 5 - 2 => a(108) = 108^3 = 1259712.

%p with(NumberTheory):

%p A363919 := n -> n^(NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct')):

%p # Alternative:

%p a := n -> local i: n^add(i[2] - 1, i in ifactors(n)[2]): seq(a(n), n = 1..68);

%t Array[#^(PrimeOmega[#] - PrimeNu[#]) &, 120]

%o (SageMath)

%o def A363919(n):

%o if n < 2: return 1

%o return n^sum(p[1] - 1 for p in list(factor(n)))

%o print([A363919(n) for n in srange(1, 69)])

%o (Julia)

%o using Nemo

%o exc(n::fmpz) = sum(e - 1 for (p, e) in factor(n))

%o A363919(n::fmpz) = n < 2 ? fmpz(1) : n^exc(n)

%o println([A363919(fmpz(n)) for n in 1:68])

%o (PARI)

%o a(n) = my(f=factor(n)[, 2]); n^(vecsum(f)-#f); \\ _Michel Marcus_, Jul 16 2023

%o (Python)

%o from sympy import factorint

%o def A363919(n): return n**sum(map(lambda e:e-1,factorint(n).values())) # _Chai Wah Wu_, Jul 18 2023

%Y Cf. A001221, A001222, A005117, A046660, A053763, A056170, A060687, A013929, A205959, A337945, A363923.

%K nonn

%O 1,4

%A _Peter Luschny_ and _Michael De Vlieger_, Jul 14 2023