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A133639 Mobius transform of b(n) where b(8n + 1) = A080995(n).

%I A133639 #14 Jan 29 2025 18:38:58
%S A133639 1,-1,-1,0,-1,1,-1,0,0,1,-1,0,-1,1,1,0,-1,0,-1,0,1,1,-1,0,1,1,0,0,-1,
%T A133639 -1,-1,0,1,1,1,0,-1,1,1,0,-1,-1,-1,0,0,1,-1,0,1,-1,1,0,-1,0,1,0,1,1,
%U A133639 -1,0,-1,1,0,0,1,-1,-1,0,1,-1,-1,0,-1,1,-1,0,1,-1,-1,0,0,1,-1,0,1,1,1,0,-1,0,1,0,1,1,1,0,-1,-1,0,0,-1,-1,-1,0,-1
%N A133639 Mobius transform of b(n) where b(8n + 1) = A080995(n).
%H A133639 Antti Karttunen, <a href="/A133639/b133639.txt">Table of n, a(n) for n = 1..10000</a>
%F A133639 a(n) is multiplicative with a(2^e) = a(3^e) = -1 if e=1, 0 if e>1, a(p^e) = (-1)^e if p > 3.
%F A133639 a(4*n) = a(9*n) = 0.
%F A133639 Dirichlet g.f.: (zeta(2*s)/zeta(s)) * (1 - 2^(-2*s)) * (1 - 3^(-2*s)). - _Amiram Eldar_, Oct 28 2023
%e A133639 G.f. = x - x^2 - x^3 - x^5 + x^6 - x^7 + x^10 - x^11 - x^13 + x^14 + x^15 + ...
%t A133639 a[ n_] := If[ n < 1, 0, Times @@ (Which[# == 1, 1, # < 5, -Boole[#2 == 1], True, (-1)^#2] & @@@ FactorInteger @ n)]; (* _Michael Somos_, Oct 31 2015 *)
%o A133639 (PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k = 1, matsize(A)[1], [p, e] = A[k, ]; if(p < 5, -(e==1), (-1)^e )))};
%o A133639 (Scheme)
%o A133639 ;; With memoization-macro definec:
%o A133639 (definec (A133639 n) (cond ((= 1 n) n) ((zero? (modulo n 4)) 0) ((zero? (modulo n 9)) 0) ((even? n) (- (A133639 (/ n 2)))) ((zero? (modulo n 3)) (- (A133639 (/ n 3)))) (else (- (A133639 (A032742 n)))))) ;; (For the code of A032742, see under that entry) _Antti Karttunen_, Sep 23 2017
%Y A133639 Cf. A080995.
%K A133639 sign,easy,mult
%O A133639 1,1
%A A133639 _Michael Somos_, Sep 14 2007