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A347092
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Dirichlet inverse of A322577, which is the convolution of Dedekind psi with Euler phi.
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3
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1, -4, -6, 5, -10, 24, -14, -4, 10, 40, -22, -30, -26, 56, 60, 5, -34, -40, -38, -50, 84, 88, -46, 24, 26, 104, -6, -70, -58, -240, -62, -4, 132, 136, 140, 50, -74, 152, 156, 40, -82, -336, -86, -110, -100, 184, -94, -30, 50, -104, 204, -130, -106, 24, 220, 56, 228, 232, -118, 300, -122, 248, -140, 5, 260, -528, -134
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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a(1) = 1; a(n) = -Sum_{d|n, d < n} A322577(n/d) * a(d).
Dirichlet g.f.: zeta(2*s)/zeta(s-1)^2.
Multiplicative with a(p^e) = 1 + p^2 if e is even, -2*p if e is odd. (End)
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MATHEMATICA
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f[p_, e_] := If[EvenQ[e], p^2 + 1, -2*p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)
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PROG
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(PARI)
up_to = 16384;
A001615(n) = if(1==n, n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
v347092 = DirInverseCorrect(vector(up_to, n, A322577(n)));
(Haskell)
import Math.NumberTheory.Primes
a n = product . map (\(p, e) -> if even e then 1 + unPrime p^2 else -2*unPrime p) . factorise $ n -- Sebastian Karlsson, Oct 29 2021
(Python)
from sympy import factorint, prod
def f(p, e): return 1 + p**2 if e%2 == 0 else -2*p
def a(n):
factors = factorint(n)
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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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