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%I A074823 #39 Jun 26 2022 02:10:35
%S A074823 1,2,2,0,2,4,2,0,0,4,2,0,2,4,4,0,2,0,2,0,4,4,2,0,0,4,0,0,2,8,2,0,4,4,
%T A074823 4,0,2,4,4,0,2,8,2,0,0,4,2,0,0,0,4,0,2,0,4,0,4,4,2,0,2,4,0,0,4,8,2,0,
%U A074823 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 A074823 a(n) = 2^omega(n)*mu(n)^2.
%H A074823 Antti Karttunen, <a href="/A074823/b074823.txt">Table of n, a(n) for n = 1..10000</a>
%H A074823 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A074823 Sum_{k=1..n} a(k) = A069201(n).
%F A074823 Multiplicative with a(p)=2, a(p^e)=0, e > 1.
%F A074823 a(n) = A034444(n)*A008966(n). - _R. J. Mathar_, Apr 15 2011
%F A074823 Sum_{n>0} a(n)/n^s = Product_{p prime} (1 + 2p^(-s)). - _Ralf Stephan_, Jul 07 2013
%F A074823 a(n) = abs(A226177(n)). - _Antti Karttunen_, Jul 23 2017
%F A074823 Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - _Vaclav Kotesovec_, Aug 20 2021
%t A074823 Table[MoebiusMu[n]^2 * 2^PrimeNu[n], {n, 1, 100}] (* _Vaclav Kotesovec_, Aug 20 2021 *)
%t A074823 f[p_, e_] :=If[e==1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jun 26 2022 *)
%o A074823 (Scheme) (define (A074823 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) 2 0) (A074823 (A028234 n))))) ;; _Antti Karttunen_, Jul 23 2017
%o A074823 (PARI) a(n) = 2^omega(n)*moebius(n)^2; \\ _Michel Marcus_, Jul 23 2017
%o A074823 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X))[n], ", ")) \\ _Vaclav Kotesovec_, Aug 20 2021
%o A074823 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^2)[n], ", ")) \\ _Vaclav Kotesovec_, Aug 20 2021
%Y A074823 Cf. A001221, A008683, A008966, A034444, A069201, A226177, A347149.
%K A074823 mult,nonn
%O A074823 1,2
%A A074823 _Benoit Cloitre_, Sep 08 2002
%E A074823 Additional comments from _Vladeta Jovovic_, Dec 30 2002

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