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A226177 a(n) = mu(n)*d(n), where mu(n) = A008683 and d(n) = A000005.

%I A226177 #42 Jun 22 2024 20:02:13
%S A226177 1,-2,-2,0,-2,4,-2,0,0,4,-2,0,-2,4,4,0,-2,0,-2,0,4,4,-2,0,0,4,0,0,-2,
%T A226177 -8,-2,0,4,4,4,0,-2,4,4,0,-2,-8,-2,0,0,4,-2,0,0,0,4,0,-2,0,4,0,4,4,-2,
%U A226177 0,-2,4,0,0,4,-8,-2,0,4,-8,-2,0,-2,4,0,0,4,-8,-2,0,0,4,-2,0,4,4,4,0,-2,0,4,0,4,4,4,0,-2,0,0,0,-2,-8,-2,0,-8
%N A226177 a(n) = mu(n)*d(n), where mu(n) = A008683 and d(n) = A000005.
%C A226177 The prime numbers are the only solutions to mu(n)*d(n) = -2.
%C A226177 Multiplicative with a(p) = -2, a(p^e) = 0, e > 1.
%C A226177 The Moebius inverse is A076479, and the Dirichlet inverse A061142. - _R. J. Mathar_, Jun 03 2013
%C A226177 Möbius transform of (-1)^omega(n). - _Wesley Ivan Hurt_, Jun 22 2024
%H A226177 Antti Karttunen, <a href="/A226177/b226177.txt">Table of n, a(n) for n = 1..10000</a>
%H A226177 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A226177 a(n) = mu(n)*d(n) = A008683(n)*A000005(n).
%F A226177 Sum_{n>0} a(n)/n^s = Product_{p prime} (1 - 2p^(-s)). - _Ralf Stephan_, Jul 07 2013
%F A226177 a(n) = mu(n) * 2^omega(n) = |mu(n)| * (-2)^omega(n), where omega = A001221. - _Álvar Ibeas_, Dec 30 2018
%F A226177 a(n) = Sum_{d|n} (-1)^omega(d) * mu(n/d). - _Wesley Ivan Hurt_, Jun 22 2024
%e A226177 a(5) = mu(5)*d(5) = (-1)(2) = -2.
%p A226177 with(numtheory); a:=n->mobius(n)*tau(n); seq(a(k),k=1..100);
%t A226177 Table[MoebiusMu[n] DivisorSigma[0, n], {n, 105}] (* _Michael De Vlieger_, Jul 23 2017 *)
%o A226177 (PARI) A226177(n) = moebius(n)*numdiv(n); \\ _Antti Karttunen_, Jul 23 2017
%o A226177 (Scheme) (define (A226177 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) -2 0) (A226177 (A028234 n))))) ;; _Antti Karttunen_, Jul 23 2017
%o A226177 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X))[n], ", ")); \\ _Vaclav Kotesovec_, Aug 21 2021
%Y A226177 Cf. A000005, A000040, A001358, A008683, A074823 (absolute values), A001221.
%K A226177 sign,mult
%O A226177 1,2
%A A226177 _Wesley Ivan Hurt_, May 29 2013
%E A226177 More terms from _Antti Karttunen_, Jul 23 2017
%E A226177 Name changed by _David A. Corneth_, Jul 23 2017