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A162511
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Multiplicative function with a(p^e) = (-1)^(e-1).
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12
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1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, -1
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OFFSET
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1,1
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LINKS
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FORMULA
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Multiplicative function with a(p^e)=(-1)^(e-1) for any prime p and any positive exponent e.
a(n) = 1 when n is a squarefree number (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A307868. - Amiram Eldar, Sep 18 2022
Dirichlet g.f.: Product_{p prime} ((p^s + 2)/(p^s + 1)). - Amiram Eldar, Oct 26 2023
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MAPLE
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local a, f;
a := 1;
for f in ifactors(n)[2] do
a := a*(-1)^(op(2, f)-1) ;
end do:
return a;
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MATHEMATICA
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PROG
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(Python)
from sympy import factorint
from operator import mul
def a(n):
f=factorint(n)
return 1 if n==1 else reduce(mul, [(-1)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017
(Python)
from functools import reduce
from sympy import factorint
def A162511(n): return -1 if reduce(lambda a, b:~(a^b), factorint(n).values(), 0)&1 else 1 # Chai Wah Wu, Jan 01 2023
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CROSSREFS
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Cf. A005117, A076479, A162510, A162512, A002035, A072587, A036537, A162643, A162644, A162645, A046660, A008836, A307868.
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KEYWORD
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easy,mult,sign
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AUTHOR
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STATUS
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approved
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