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A318310 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A033879(i) = A033879(j), for all i, j >= 0.

%I A318310 #20 Jan 14 2020 09:59:54
%S A318310 1,1,2,1,3,4,5,1,6,2,7,8,9,10,11,1,12,13,14,15,7,16,17,18,19,7,20,21,
%T A318310 22,23,24,1,25,26,27,28,29,30,17,31,32,33,34,10,35,36,37,38,39,40,41,
%U A318310 5,42,23,43,44,45,46,47,48,49,50,51,1,52,18,53,54,55,56,57,58,59,60,46,9,61,23,62,63,39,64,65,66,67,68,69,56,70,71,72,73,47,74
%N A318310 Lexicographically earliest infinite sequence such that a(i) = a(j) => A000120(i) = A000120(j) and A033879(i) = A033879(j), for all i, j >= 0.
%C A318310 Restricted growth sequence transform of ordered pair [A000120(n), A033879(n)], or equally, of ordered pair [A000120(n), A294898(n)].
%C A318310 For all i, j:
%C A318310   A318311(i) = A318311(j) => a(i) = a(j),
%C A318310   a(i) = a(j) => A286449(i) = A286449(j),
%C A318310   a(i) = a(j) => A294898(i) = A294898(j).
%C A318310 In the scatter plot one can see the effects of both base-2 related A000120 (binary weight of n) and prime factorization related A033879 (deficiency of n) graphically mixed: from the former, a square grid pattern, and from the latter the black rays that emanate from the origin. The same is true for A323898, while in the ordinal transform of this sequence, A331184, such effects are harder to visually discern. - _Antti Karttunen_, Jan 13 2020
%H A318310 Antti Karttunen, <a href="/A318310/b318310.txt">Table of n, a(n) for n = 1..65537</a>
%o A318310 (PARI)
%o A318310 up_to = 65537;
%o A318310 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 A318310 A318310aux(n) = [hammingweight(n), (2*n) - sigma(n)];
%o A318310 v318310 = rgs_transform(vector(up_to,n,A318310aux(n)));
%o A318310 A318310(n) = v318310[n];
%Y A318310 Cf. A000120, A033879, A286449, A295881, A295882, A294898.
%Y A318310 Cf. A318311, A323889, A323892, A323898, A324344, A324380, A324390 for similar constructions.
%Y A318310 Cf. A331184 (ordinal transform).
%K A318310 nonn,look
%O A318310 1,3
%A A318310 _Antti Karttunen_, Aug 25 2018
%E A318310 Name changed by _Antti Karttunen_, Jan 13 2020