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A344025 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A003557(i) = A003557(j), for all i, j >= 1.

%I A344025 #14 Jan 15 2024 19:18:54
%S A344025 1,2,2,3,2,4,2,5,6,7,2,8,2,9,10,11,2,12,2,13,14,15,2,16,17,18,19,20,2,
%T A344025 21,2,22,23,24,25,26,2,27,28,29,2,30,2,31,32,33,2,34,35,36,37,38,2,39,
%U A344025 28,40,41,21,2,42,2,43,44,45,46,47,2,48,49,50,2,51,2,52,53,54,46,55,2,56,57,58,2,59,41,60,61,62,2,63,37,64
%N A344025 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A003557(i) = A003557(j), for all i, j >= 1.
%C A344025 Restricted growth sequence transform of the ordered pair [A003415(n), A003557(n)], where A003415(n) is the arithmetic derivative of n, and A003557(n) is n divided by its largest squarefree divisor.
%C A344025 For all i, j:
%C A344025   parent(i) = parent(j) => a(i) = a(j),
%C A344025   a(i) = a(j) => A342001(i) = A342001(j),
%C A344025   a(i) = a(j) => A369051(i) = A369051(j) => A085731(i) = A085731(j).
%C A344025 Where "parent" can be any of the sequences A351236, A351260, A353520, A353521, A369050, for example.
%H A344025 Antti Karttunen, <a href="/A344025/b344025.txt">Table of n, a(n) for n = 1..65537</a>
%o A344025 (PARI)
%o A344025 up_to = 65537;
%o A344025 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 A344025 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A344025 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A344025 Aux344025(n) = [A003415(n), A003557(n)];
%o A344025 v344025 = rgs_transform(vector(up_to, n, Aux344025(n)));
%o A344025 A344025(n) = v344025[n];
%Y A344025 Cf. A003415, A003557, A085731, A342001, A369051.
%Y A344025 Cf. also A305800, A351236, A351260, A353520, A353521, A369050, and also A353523.
%K A344025 nonn
%O A344025 1,2
%A A344025 _Antti Karttunen_, May 07 2021