A318310 - OEIS

%I #20 Jan 14 2020 09:59:54

%S 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 22,23,24,1,25,26,27,28,29,30,17,31,32,33,34,10,35,36,37,38,39,40,41,

%U 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 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 Restricted growth sequence transform of ordered pair [A000120(n), A033879(n)], or equally, of ordered pair [A000120(n), A294898(n)].

%C For all i, j:

%C   A318311(i) = A318311(j) => a(i) = a(j),

%C   a(i) = a(j) => A286449(i) = A286449(j),

%C   a(i) = a(j) => A294898(i) = A294898(j).

%C 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 Antti Karttunen, <a href="/A318310/b318310.txt">Table of n, a(n) for n = 1..65537</a>

%o (PARI)

%o up_to = 65537;

%o 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 A318310aux(n) = [hammingweight(n), (2*n) - sigma(n)];

%o v318310 = rgs_transform(vector(up_to,n,A318310aux(n)));

%o A318310(n) = v318310[n];

%Y Cf. A000120, A033879, A286449, A295881, A295882, A294898.

%Y Cf. A318311, A323889, A323892, A323898, A324344, A324380, A324390 for similar constructions.

%Y Cf. A331184 (ordinal transform).

%K nonn,look

%O 1,3

%A _Antti Karttunen_, Aug 25 2018

%E Name changed by _Antti Karttunen_, Jan 13 2020