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A358272
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Multiplicative sequence with a(p^e) = (-1)^e * p^(2*floor(e/2)) for prime p and e >= 0.
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1
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1, -1, -1, 4, -1, 1, -1, -4, 9, 1, -1, -4, -1, 1, 1, 16, -1, -9, -1, -4, 1, 1, -1, 4, 25, 1, -9, -4, -1, -1, -1, -16, 1, 1, 1, 36, -1, 1, 1, 4, -1, -1, -1, -4, -9, 1, -1, -16, 49, -25, 1, -4, -1, 9, 1, 4, 1, 1, -1, 4, -1, 1, -9, 64, 1, -1, -1, -4, 1, -1, -1, -36, -1, 1, -25, -4, 1, -1, -1, -16
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OFFSET
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1,4
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COMMENTS
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(2*s-2) / zeta(s).
Dirichlet inverse b(n), n > 0, is multiplicative with b(p) = 1 and b(p^e) = 1 - p^2 for prime p and e > 1.
Conjecture: a(n) = Sum_{k=1..n} gcd(k, n) * lambda(gcd(k, n)) for n > 0.
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MAPLE
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local a, pe, e, p ;
a := 1;
for pe in ifactors(n)[2] do
e := op(2, pe) ;
p := op(1, pe) ;
a := a*(-1)^e*p^(2*floor(e/2)) ;
end do:
a ;
end proc:
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MATHEMATICA
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f[p_, e_] := (-1)^e * p^(2*Floor[e/2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 07 2022 *)
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PROG
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(Python)
from math import prod
from sympy import factorint
def A358272(n): return prod(-p**(e&-2) if e&1 else p**(e&-2) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 17 2023
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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