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A358272 Multiplicative sequence with a(p^e) = (-1)^e * p^(2*floor(e/2)) for prime p and e >= 0.

%I A358272 #18 May 14 2025 00:27:15
%S A358272 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,
%T A358272 -9,-4,-1,-1,-1,-16,1,1,1,36,-1,1,1,4,-1,-1,-1,-4,-9,1,-1,-16,49,-25,
%U A358272 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
%N A358272 Multiplicative sequence with a(p^e) = (-1)^e * p^(2*floor(e/2)) for prime p and e >= 0.
%C A358272 Signed version of A008833.
%F A358272 a(n) = lambda(n) * A008833(n) for n > 0 where lambda(n) = A008836(n).
%F A358272 Dirichlet g.f.: zeta(2*s-2) / zeta(s).
%F A358272 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.
%F A358272 Dirichlet convolution with A034444 equals A008833.
%F A358272 Equals Dirichlet convolution of A000010 and A061019.
%F A358272 Conjecture: a(n) = Sum_{k=1..n} gcd(k, n) * lambda(gcd(k, n)) for n > 0.
%F A358272 a(n) = Sum_{d|n} lambda(d)*d*phi(n/d), where lambda(n) = A008836(n). - _Ridouane Oudra_, May 11 2025
%p A358272 A358272 := proc(n)
%p A358272     local a,pe,e,p ;
%p A358272     a := 1;
%p A358272     for pe in ifactors(n)[2] do
%p A358272         e := op(2,pe) ;
%p A358272         p := op(1,pe) ;
%p A358272         a := a*(-1)^e*p^(2*floor(e/2)) ;
%p A358272     end do:
%p A358272     a ;
%p A358272 end proc:
%p A358272 seq(A358272(n),n=1..80) ; # _R. J. Mathar_, Jan 17 2023
%t A358272 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 *)
%o A358272 (Python)
%o A358272 from math import prod
%o A358272 from sympy import factorint
%o A358272 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
%Y A358272 Cf. A000010, A008833, A008836, A034444, A061019.
%K A358272 sign,easy,mult
%O A358272 1,4
%A A358272 _Werner Schulte_, Nov 07 2022