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A353493 The arithmetic derivative of n, reduced modulo 4.

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%I A353493 #19 Apr 27 2022 13:55:23
%S A353493 0,0,1,1,0,1,1,1,0,2,3,1,0,1,1,0,0,1,1,1,0,2,1,1,0,2,3,3,0,1,3,1,0,2,
%T A353493 3,0,0,1,1,0,0,1,1,1,0,3,1,1,0,2,1,0,0,1,1,0,0,2,3,1,0,1,1,3,0,2,1,1,
%U A353493 0,2,3,1,0,1,3,3,0,2,3,1,0,0,3,1,0,2,1,0,0,1,3,0,0,2,1,0,0,1,1,3,0,1,3,1,0,3
%N A353493 The arithmetic derivative of n, reduced modulo 4.
%H A353493 Antti Karttunen, <a href="/A353493/b353493.txt">Table of n, a(n) for n = 0..65537</a>
%F A353493 a(n) = A010873(A003415(n)).
%F A353493 For all n, a(4*n) = 0 and a(4*n + 2) is either 1 or 3. [See comments in A235991]
%F A353493 For all n >= 2, a(n) = A010873[(A353496(n)*A353497(n)) + A353490(n)]. (This is essentially _Reinhard Zumkeller_'s May 09 2011 recursive formula of A003415, when reduced modulo 4) - _Antti Karttunen_, Apr 26 2022
%o A353493 (PARI)
%o A353493 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A353493 A353493(n) = (A003415(n)%4);
%Y A353493 Cf. A003415, A010873, A165560, A235991, A353490, A353494, A353495, A353496, A353497.
%K A353493 nonn
%O A353493 0,10
%A A353493 _Antti Karttunen_, Apr 22 2022