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Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.
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%I #15 Feb 14 2022 11:22:17

%S 1,2,3,2,4,3,5,2,6,4,7,3,8,5,9,2,10,6,11,4,12,7,13,3,14,8,15,5,16,9,

%T 17,2,18,10,19,6,20,11,21,4,22,12,23,7,24,13,25,3,26,14,27,8,28,15,29,

%U 5,30,16,31,9,32,17,33,2,34,18,35,10,36,19,37,6,38,20,39,11,40,21,41,4,42,22,43,12,44,23,45,7,46,24,47,13,48,25,49,3,50,26,51,14,52,27,53,8,54

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A006530(i) = A006530(j), A206787(i) = A206787(j) and A336651(i) = A336651(j) for all i, j >= 1.

%C Restricted growth sequence transform of the ordered triplet [A006530(n), A206787(n), A336651(n)].

%C For all i, j >= 1:

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

%C a(i) = a(j) => A347241(i) = A347241(j),

%C a(i) = a(j) => A351461(i) = A351461(j) => A347240(i) = A347240(j).

%H Antti Karttunen, <a href="/A351460/b351460.txt">Table of n, a(n) for n = 1..65537</a>

%e a(429) = a(455) because 429 = 3*11*13 and 455 = 5*7*13, so they have equal largest prime factor (A006530), and they also agree on A206787(429) = A206787(455) = 672 and on A336651(429) = A336651(455) = 1 (because both are squarefree), therefore they get equal value (which is 216) allotted to them by the restricted growth sequence transform. - _Antti Karttunen_, Feb 14 2022

%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 A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);

%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 Aux351460(n) = [A006530(n), A206787(n), A336651(n)];

%o v351460 = rgs_transform(vector(up_to, n, Aux351460(n)));

%o A351460(n) = v351460[n];

%Y Cf. A006530, A206787, A336651, A347241, A351461.

%Y Cf. also A324400, A351452.

%Y Differs from A351454 for the first time at n=121, where a(121) = 62, while A351454(121) = 51.

%Y Differs from A103391(1+n) for the first time after n=1 at n=455, where a(455) = 216, while A103391(456) = 229.

%K nonn

%O 1,2

%A _Antti Karttunen_, Feb 11 2022