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A324345 Lexicographically earliest positive sequence such that a(i) = a(j) => A005811(i) = A005811(j) and A278222(i) = A278222(j), for all i, j >= 0.

%I A324345 #10 Feb 24 2019 17:52:32
%S A324345 1,2,3,4,3,5,6,7,3,5,8,9,6,9,10,11,3,5,8,9,8,12,13,14,6,9,13,15,10,14,
%T A324345 16,17,3,5,8,9,8,12,13,14,8,12,18,19,13,19,20,21,6,9,13,15,13,19,22,
%U A324345 23,10,14,20,23,16,21,24,25,3,5,8,9,8,12,13,14,8,12,18,19,13,19,20,21,8,12,18,19,18,26,27,28,13,19,27,29,20,28,30,31,6,9,13,15,13,19
%N A324345 Lexicographically earliest positive sequence such that a(i) = a(j) => A005811(i) = A005811(j) and A278222(i) = A278222(j), for all i, j >= 0.
%C A324345 Restricted growth sequence transform of the ordered pair [A005811(n), A278222(n)], or equally, of [A005811(n), A286622(n)].
%C A324345 For all i, j >= 1:
%C A324345   a(i) = a(j) => A033264(i) = A033264(j).
%H A324345 Antti Karttunen, <a href="/A324345/b324345.txt">Table of n, a(n) for n = 0..65537</a>
%H A324345 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A324345 a(2^n) = 3 for all n >= 1.
%o A324345 (PARI)
%o A324345 up_to = 65537;
%o A324345 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 A324345 A005811(n) = hammingweight(bitxor(n, n>>1)); \\ From A005811
%o A324345 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
%o A324345 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A324345 A278222(n) = A046523(A005940(1+n));
%o A324345 Aux324345(n) = [A005811(n), A278222(n)];
%o A324345 v324345 = rgs_transform(vector(1+up_to,n,Aux324345(n-1)));
%o A324345 A324345(n) = v324345[1+n];
%Y A324345 Cf. A005811, A278222, A286622.
%Y A324345 Cf. also A033264, A323889, A324343, A324346.
%K A324345 nonn
%O A324345 0,2
%A A324345 _Antti Karttunen_, Feb 24 2019