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Lexicographically earliest infinite sequence such that a(i) = a(j) => A369066(i) = A369066(j) for all i, j >= 0.
4

%I #9 Jan 16 2024 16:51:10

%S 1,2,2,1,2,3,2,3,2,4,5,1,2,6,1,7,2,8,9,1,10,11,4,9,2,6,3,1,6,1,12,5,2,

%T 13,14,1,11,15,8,11,16,17,18,1,8,19,1,20,2,6,3,1,4,4,6,4,6,5,9,1,6,8,

%U 2,20,2,21,11,1,16,22,13,15,23,24,25,1,13,21,1,26,27,28,29,1,30,29,17,31,13,21,15,1,16

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A369066(i) = A369066(j) for all i, j >= 0.

%C Restricted growth sequence transform of A369066.

%H Antti Karttunen, <a href="/A369067/b369067.txt">Table of n, a(n) for n = 0..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 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };

%o A008836(n) = ((-1)^bigomega(n));

%o A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

%o A369069(n) = sumdiv(n,d,A008836(n/d)*A083345(d));

%o A369066(n) = A369069(A005940(1+n));

%o v369067 = rgs_transform(vector(1+up_to,n,A369066(n-1)));

%o A369067(n) = v369067[1+n];

%Y Cf. A005940, A008836, A083345, A369066, A369069.

%Y Cf. also A366805 (compare the scatter plots).

%K nonn

%O 0,2

%A _Antti Karttunen_, Jan 16 2024