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A369046 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A069359(i) = A069359(j), for all i, j >= 1.

%I A369046 #8 Jan 18 2024 17:57:06
%S A369046 1,2,2,3,2,4,2,5,6,7,2,8,2,9,10,11,2,12,2,13,14,15,2,16,17,18,19,20,2,
%T A369046 21,2,22,23,24,25,26,2,27,28,29,2,30,2,31,32,33,2,34,35,36,37,38,2,39,
%U A369046 28,40,41,21,2,42,2,43,44,45,46,47,2,48,49,50,2,51,2,52,53,54,46,55,2,56,57,58,2,59,41,60,61,62,2,63,37
%N A369046 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j) and A069359(i) = A069359(j), for all i, j >= 1.
%C A369046 Restricted growth sequence transform of the ordered pair [A003415(n), A069359(n)].
%C A369046 For all i,j >= 1:
%C A369046   A369050(i) = A369050(j) => a(i) = a(j), [Conjectured]
%C A369046   a(i) = a(j) => A329039(i) = A329039(j) => A008966(i) = A008966(j).
%H A369046 Antti Karttunen, <a href="/A369046/b369046.txt">Table of n, a(n) for n = 1..65537</a>
%o A369046 (PARI)
%o A369046 up_to = 65537;
%o A369046 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 A369046 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369046 A069359(n) = (n*sumdiv(n, d, isprime(d)/d));
%o A369046 Aux369046(n) = [A003415(n), A069359(n)];
%o A369046 v369046 = rgs_transform(vector(up_to, n, Aux369046(n)));
%o A369046 A369046(n) = v369046[n];
%Y A369046 Cf. A003415, A008966, A069359, A329039.
%Y A369046 Cf. also A369050, A369051.
%Y A369046 Differs from A351236 for the first time at n=91, where a(91) = 37, while A351236(91) = 64.
%Y A369046 Differs from A344025 for the first time at n=140, where a(140) = 97, while A344025(140) = 92.
%K A369046 nonn
%O A369046 1,2
%A A369046 _Antti Karttunen_, Jan 18 2024