A363919 a(n) = n^excess(n), where excess(n) = A046660(n).

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, 1, 1, 1, 1048576, 1, 1, 1, 1296, 1, 1, 1, 1600, 1, 1, 1, 44, 45, 1, 1, 110592, 49, 50, 1, 52, 1, 2916, 1, 3136, 1, 1, 1, 60, 1, 1, 63, 1073741824, 1, 1, 1, 68

Offset: 1

Author

Peter Luschny and Michael De Vlieger, Jul 14 2023

Keywords

• nonn

Examples

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

Links

• Plot2 A363923 vs A205959.
• Michael De Vlieger, Plot p(k)^e(k) at (x,y) = (n,k) 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.
• Michael De Vlieger, Notes on certain sequences relating BigOmega(n), omega(n), and rad(n), 2023.

Crossrefs

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

Programs

Julia

using Nemo
exc(n::fmpz) = sum(e - 1 for (p, e) in factor(n))
A363919(n::fmpz) = n < 2 ? fmpz(1) : n^exc(n)
println([A363919(fmpz(n)) for n in 1:68])

Maple

with(NumberTheory):
A363919 := n -> n^(NumberOfPrimeFactors(n) - NumberOfPrimeFactors(n, 'distinct')):
# Alternative:
a := n -> local i: n^add(i[2] - 1, i in ifactors(n)[2]): seq(a(n), n = 1..68);

Mathematica

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

PARI

a(n) = my(f=factor(n)[, 2]); n^(vecsum(f)-#f); \\ Michel Marcus, Jul 16 2023

Python

from sympy import factorint
def A363919(n): return n**sum(map(lambda e:e-1,factorint(n).values())) # Chai Wah Wu, Jul 18 2023

SageMath

def A363919(n):
    if n < 2: return 1
    return n^sum(p[1] - 1 for p in list(factor(n)))
print([A363919(n) for n in srange(1, 69)])

Formula

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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