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A365718 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365717(i) = A365717(j) for all i, j >= 0, where A365717(n) = A348717(A356867(1+n)).

%I A365718 #19 Sep 17 2023 13:58:08
%S A365718 1,2,2,2,3,4,5,6,3,2,7,4,3,8,9,10,11,12,4,13,14,6,15,16,17,18,6,2,19,
%T A365718 5,5,20,21,22,23,9,3,24,12,9,25,26,27,28,29,12,30,31,11,32,33,34,35,
%U A365718 36,4,37,14,8,38,39,40,41,42,6,43,36,16,44,45,46,47,48,29,49,50,18,51,52,53,54,11,2,55,7,7,56
%N A365718 Lexicographically earliest infinite sequence such that a(i) = a(j) => A365717(i) = A365717(j) for all i, j >= 0, where A365717(n) = A348717(A356867(1+n)).
%C A365718 Restricted growth sequence transform of A365717.
%C A365718 For all i, j >= 0: a(i) = a(j) => A365720(i) = A365720(j).
%C A365718 In contrast to austere A103391, which is easily computed from n's binary expansion, the scatter plot here with its slender seaweed-like branchings suggests that this sequence is not just a simple derivation of base-3 expansion of n.
%H A365718 Antti Karttunen, <a href="/A365718/b365718.txt">Table of n, a(n) for n = 0..59049</a>
%o A365718 (PARI)
%o A365718 up_to = 59049; \\ = 3^10.
%o A365718 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 A365718 A348717(n) = if(1==n, 1, my(f = factor(n), k = primepi(f[1, 1])-1); for (i=1, #f~, f[i, 1] = prime(primepi(f[i, 1])-k)); factorback(f));
%o A365718 A356867list(up_to) = { my(v=vector(up_to),met=Map(),h=0,ak); for(i=1,#v,if(1==vecsum(digits(i,3)), v[i] = i; h = i, ak = v[i-h]; forprime(p=2,,if(3!=p && !mapisdefined(met,p*ak), v[i] = p*ak; break))); mapput(met,v[i],i)); (v); };
%o A365718 v365718 = rgs_transform(apply(A348717,A356867list(1+up_to)));
%o A365718 A365718(n) = v365718[1+n];
%Y A365718 Cf. A348717, A356867, A365720.
%Y A365718 Cf. also A103391 (similar transformation applied to A005940) and A365715 (compare the scatter plot).
%K A365718 nonn,look
%O A365718 0,2
%A A365718 _Antti Karttunen_, Sep 17 2023