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Lexicographically earliest infinite sequence such that a(i) = a(j) => A344026(i) = A344026(j) for all i, j >= 0.
3

%I #10 Jan 16 2024 16:51:14

%S 1,2,2,3,2,4,5,6,2,7,8,9,10,11,12,13,2,14,10,15,6,16,17,18,19,20,21,

%T 22,23,24,25,26,2,27,19,13,9,28,29,30,31,32,33,34,35,36,37,38,39,40,

%U 41,42,43,44,45,46,47,48,49,50,51,52,53,54,2,55,9,56,31,57,23,34,58,59,60,61,62,63,50,64,15,65,66

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A344026(i) = A344026(j) for all i, j >= 0.

%C Restricted growth sequence transform of A344026, i.e., of the arithmetic derivative (A003415) as reordered by the Doudna sequence (A005940).

%H Antti Karttunen, <a href="/A369065/b369065.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 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };

%o A344026(n) = A003415(A005940(1+n));

%o v369065 = rgs_transform(vector(1+up_to,n,A344026(n-1)));

%o A369065(n) = v369065[1+n];

%Y Cf. A003415, A005940, A344026.

%Y Cf. also A366802, A369067.

%K nonn

%O 0,2

%A _Antti Karttunen_, Jan 16 2024