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A353523 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349905(i) = A349905(j) and A003557(i) = A003557(j), for all i, j >= 1.

%I A353523 #12 Apr 28 2022 13:34:36
%S A353523 1,2,2,3,2,4,2,5,6,7,2,8,2,9,10,11,2,12,2,13,14,14,2,15,16,17,18,19,2,
%T A353523 20,2,21,22,23,22,24,2,25,23,26,2,27,2,28,29,30,2,31,32,33,34,35,2,36,
%U A353523 17,37,38,39,2,40,2,41,42,43,34,44,2,45,39,46,2,47,2,48,49,50,34,51,2,52,53,54,2,55,25,56,57,58,2,59,38,60
%N A353523 Lexicographically earliest infinite sequence such that a(i) = a(j) => A349905(i) = A349905(j) and A003557(i) = A003557(j), for all i, j >= 1.
%C A353523 Restricted growth sequence transform of the ordered pair [A003557(n), A349905(n)], or equally, of the ordered pair [A003415(A003961(n)), A003557(A003961(n))].
%C A353523 This is a prime-shifted variant of A344025, as this is the restricted growth sequence transform of A344025(A003961(n)).
%C A353523 For all i, j:
%C A353523   A305800(i) = A305800(j) => a(i) = a(j),
%C A353523   a(i) = a(j) => A349905(i) = A349905(j) => A008836(i) = A008836(j),
%C A353523   a(i) = a(j) => A353571(i) = A353571(j).
%H A353523 Antti Karttunen, <a href="/A353523/b353523.txt">Table of n, a(n) for n = 1..65537</a>
%H A353523 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%o A353523 (PARI)
%o A353523 up_to = 65537;
%o A353523 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 A353523 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A353523 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A353523 A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
%o A353523 Aux353523(n) = { my(s=A003961(n)); [A003415(s), A003557(s)]; };
%o A353523 v353523 = rgs_transform(vector(up_to, n, Aux353523(n)));
%o A353523 A353523(n) = v353523[n];
%Y A353523 Cf. A003415, A003557, A003961, A008836, A305800, A344025, A349905, A353571.
%K A353523 nonn
%O A353523 1,2
%A A353523 _Antti Karttunen_, Apr 27 2022