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A363919
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a(n) = n^excess(n), where excess(n) = A046660(n).
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3
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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
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OFFSET
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1,4
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LINKS
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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.
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FORMULA
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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) = 1 or divisible by at least one squared prime.
a(n) = 1 <=> n is squarefree (A005117).
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) > 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.
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EXAMPLE
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108 = 2^2 * 3^3 => excess(108) = 5 - 2 => a(108) = 108^3 = 1259712.
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MAPLE
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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);
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MATHEMATICA
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Array[#^(PrimeOmega[#] - PrimeNu[#]) &, 120]
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PROG
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(SageMath)
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)])
(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])
(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
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CROSSREFS
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Cf. A001221, A001222, A005117, A046660, A053763, A056170, A060687, A013929, A205959, A337945, A363923.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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