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%I A080736 #18 Nov 02 2023 04:18:10
%S A080736 1,0,2,2,4,0,6,4,6,0,10,4,12,0,8,8,16,0,18,8,12,0,22,8,20,0,18,12,28,
%T A080736 0,30,16,20,0,24,12,36,0,24,16,40,0,42,20,24,0,46,16,42,0,32,24,52,0,
%U A080736 40,24,36,0,58,16,60,0,36,32,48,0,66,32,44,0,70,24,72,0,40,36,60,0,78,32
%N A080736 Multiplicative function defined by a(1)=1, a(2)=0, a(2^r) = phi(2^r) (r>1), a(p^r) = phi(p^r) (p odd prime, r>=1), where phi is Euler's function A000010.
%H A080736 Reinhard Zumkeller, <a href="/A080736/b080736.txt">Table of n, a(n) for n = 1..10000</a>
%F A080736 a(A016825(n)) = 0; a(A000040(n)) = A000040(n) - 1. - _Reinhard Zumkeller_, Jun 11 2012
%F A080736 a(n) = if n mod 4 = 2 then 0 else A000010(n). - _Reinhard Zumkeller_, Jun 13 2012
%F A080736 From _Amiram Eldar_, Nov 02 2023: (Start)
%F A080736 Multiplicative with a(2) = 0, a(2^e) = 2^(e-1) for e >= 2, and a(p^e) = (p-1)*p^(e-1) for an odd prime p.
%F A080736 Dirichlet g.f.: (1 - 2^(1-s) + 1/(2^s-1)) * zeta(s-1) / zeta(s).
%F A080736 Sum_{k=1..n} a(k) ~ (5/(2*Pi^2)) * n^2. (End)
%t A080736 a[n_] := If[Mod[n, 4] == 2, 0, EulerPhi[n]]; Array[a, 100] (* _Amiram Eldar_, Nov 02 2023 *)
%o A080736 (PARI) {for(n=1,81,f=factor(n); print1(if(n==1,1,if(f[1,1]==2&&f[1,2]==1,0,prod(j=1,matsize(f)[1],eulerphi(f[j,1]^f[j,2])))),","))}
%o A080736 (Haskell)
%o A080736 a080736 n = if n `mod` 4 == 2 then 0 else a000010 n
%o A080736 -- _Reinhard Zumkeller_, Jun 13 2012, Jun 11 2012
%Y A080736 Cf. A080737.
%Y A080736 Cf. A000010, A141809.
%K A080736 nonn,easy,mult
%O A080736 1,3
%A A080736 _N. J. A. Sloane_, Mar 08 2003
%E A080736 More terms and PARI code from _Klaus Brockhaus_, Mar 10 2003

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