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A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

%I A351461 #23 Nov 23 2023 15:14:03
%S A351461 1,1,2,1,3,2,4,1,5,3,6,2,7,4,8,1,9,5,10,3,11,6,8,2,12,7,13,4,14,8,11,
%T A351461 1,15,9,15,5,16,10,17,3,18,11,19,6,20,8,15,2,21,12,22,7,23,13,22,4,24,
%U A351461 14,25,8,26,11,27,1,28,15,29,9,30,15,22,5,31,16,32,10,30,17,24,3,33,18,28,11,34,19,35,6,36,20,37,8,38,15,35,2,39,21,40,12,41,22,42,7,43
%N A351461 Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.
%C A351461 Restricted growth sequence transform of the ordered pair [A206787(n), A336651(n)], or equally, of sequence b(n) = A291750(A000265(n)).
%C A351461 For all i, j >= 1:
%C A351461   A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),
%C A351461   A324400(i) = A324400(j) => A351460(i) = A351460(j) => a(i) = a(j),
%C A351461   a(i) = a(j) => A000593(i) = A000593(j),
%C A351461   a(i) = a(j) => A347385(i) = A347385(j),
%C A351461   a(i) = a(j) => A351037(i) = A351037(j) => A347240(i) = A347240(j).
%C A351461 From _Antti Karttunen_, Nov 23 2023: (Start)
%C A351461 Conjectured to be equal to the lexicographically earliest infinite sequence such that b(i) = b(j) => A000593(i) = A000593(j) and A336467(i) = A336467(j) for all i, j >= 1. In any case, a(i) = a(j) => b(i) = b(j) for all i, j >= 1 [because both A000593(n) and A336467(n) can be computed from the values of A206787(n) and A336651(n)], but whether the implication holds to the opposite direction is still open. Empirically this has been checked up to n = 2^22. See also comment in A351040.
%C A351461 (End)
%H A351461 Antti Karttunen, <a href="/A351461/b351461.txt">Table of n, a(n) for n = 1..65537</a>
%o A351461 (PARI)
%o A351461 up_to = 65537;
%o A351461 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 A351461 A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ From A206787
%o A351461 A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };
%o A351461 Aux351461(n) = [A206787(n), A336651(n)];
%o A351461 v351461 = rgs_transform(vector(up_to, n, Aux351461(n)));
%o A351461 A351461(n) = v351461[n];
%Y A351461 Cf. A206787, A336651.
%Y A351461 Cf. also A000593, A003602, A291750, A291751, A324400, A336467, A347240, A347385, A351040, A351460.
%Y A351461 Differs from A351037 for the first time at n=103, where a(103) = 42 while A351037(103) = 27.
%K A351461 nonn
%O A351461 1,3
%A A351461 _Antti Karttunen_, Feb 11 2022