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A369051 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A085731(i) = A085731(j), for all i, j >= 1.

%I A369051 #7 Jan 15 2024 19:19:12
%S A369051 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 A369051 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 A369051 28,40,41,21,2,40,2,42,43,44,45,46,2,47,48,49,2,50,2,51,52,53,45,54,2,55,56,57,2,58,41,59,60,61,2,62,37
%N A369051 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A085731(i) = A085731(j), for all i, j >= 1.
%C A369051 Restricted growth sequence transform of the ordered pair [A003415(n), A085731(n)], or equally, of the pair [A003415(n), A083345(n)], or equally, of the pair [A083345(n), A085731(n)].
%C A369051 For all i, j: A369050(i) = A369050(j) => A344025(i) = A344025(j) => a(i) = a(j).
%H A369051 Antti Karttunen, <a href="/A369051/b369051.txt">Table of n, a(n) for n = 1..65537</a>
%o A369051 (PARI)
%o A369051 up_to = 65537;
%o A369051 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 A369051 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369051 A085731(n) = gcd(A003415(n),n);
%o A369051 Aux369051(n) = [A003415(n), A085731(n)];
%o A369051 v369051 = rgs_transform(vector(up_to, n, Aux369051(n)));
%o A369051 A369051(n) = v369051[n];
%Y A369051 Cf. A003415, A083345, A085731.
%Y A369051 Cf. also A344025, A369050.
%K A369051 nonn
%O A369051 1,2
%A A369051 _Antti Karttunen_, Jan 15 2024