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Lexicographically earliest infinite sequence such that a(i) = a(j) => A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.
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%I #23 Nov 23 2023 15:14:03

%S 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 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 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 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 Restricted growth sequence transform of the ordered pair [A206787(n), A336651(n)], or equally, of sequence b(n) = A291750(A000265(n)).

%C For all i, j >= 1:

%C A003602(i) = A003602(j) => A351040(i) = A351040(j) => a(i) = a(j),

%C A324400(i) = A324400(j) => A351460(i) = A351460(j) => a(i) = a(j),

%C a(i) = a(j) => A000593(i) = A000593(j),

%C a(i) = a(j) => A347385(i) = A347385(j),

%C a(i) = a(j) => A351037(i) = A351037(j) => A347240(i) = A347240(j).

%C From _Antti Karttunen_, Nov 23 2023: (Start)

%C 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 (End)

%H Antti Karttunen, <a href="/A351461/b351461.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 A206787(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ From A206787

%o A336651(n) = { my(f=factor(n)); prod(i=1, #f~, if(2==f[i,1],1,f[i,1]^(f[i,2]-1))); };

%o Aux351461(n) = [A206787(n), A336651(n)];

%o v351461 = rgs_transform(vector(up_to, n, Aux351461(n)));

%o A351461(n) = v351461[n];

%Y Cf. A206787, A336651.

%Y Cf. also A000593, A003602, A291750, A291751, A324400, A336467, A347240, A347385, A351040, A351460.

%Y Differs from A351037 for the first time at n=103, where a(103) = 42 while A351037(103) = 27.

%K nonn

%O 1,3

%A _Antti Karttunen_, Feb 11 2022