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A353522 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A003415(i) = A003415(j), for all i, j >= 1, where A000035 and A003415 compute the parity and the arithmetic derivative of their argument.

%I A353522 #10 Apr 27 2022 20:01:32
%S A353522 1,2,3,4,3,5,3,6,7,8,3,9,3,10,11,12,3,13,3,14,15,16,3,17,15,18,19,12,
%T A353522 3,20,3,21,22,23,24,25,3,13,26,27,3,28,3,29,30,31,3,32,22,33,34,35,3,
%U A353522 36,26,37,38,20,3,37,3,39,40,41,42,43,3,44,45,46,3,47,3,48,49,21,42,50,3,51,52,53,3,54,38,33,55,56,3,57,34,58
%N A353522 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000035(i) = A000035(j) and A003415(i) = A003415(j), for all i, j >= 1, where A000035 and A003415 compute the parity and the arithmetic derivative of their argument.
%C A353522 Restricted growth sequence transform of the ordered pair [A000035(n), A003415(n)].
%C A353522 For all i, j:
%C A353522    A353520(i) = A353520(j) => A353521(i) = A353521(j) => a(i) = a(j).
%H A353522 Antti Karttunen, <a href="/A353522/b353522.txt">Table of n, a(n) for n = 1..65537</a>
%o A353522 (PARI)
%o A353522 up_to = 65537;
%o A353522 rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
%o A353522 A000035(n) = (n%2);
%o A353522 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A353522 Aux353522(n) = [A000035(n), A003415(n)];
%o A353522 v353522 = rgs_transform(vector(up_to,n,Aux353522(n)));
%o A353522 A353522(n) = v353522[n];
%Y A353522 Cf. A000035, A003415.
%Y A353522 Cf. also A305801, A353520, A353521.
%K A353522 nonn
%O A353522 1,2
%A A353522 _Antti Karttunen_, Apr 27 2022