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A369448 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A327858(i) = A327858(j) and A359589(i) = A359589(j), for all i, j >= 1.

%I A369448 #7 Jan 26 2024 10:18:20
%S A369448 1,2,2,3,2,4,2,5,6,7,2,8,2,9,10,11,2,12,2,13,14,15,2,16,14,17,18,11,2,
%T A369448 19,2,20,21,22,23,24,2,25,26,27,2,28,2,29,30,31,2,32,21,33,34,35,2,36,
%U A369448 26,37,38,19,2,37,2,39,40,41,42,43,2,44,45,46,2,47,2,30,48,49,42,50,2,51,52,53,2,54,38,33,55
%N A369448 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A327858(i) = A327858(j) and A359589(i) = A359589(j), for all i, j >= 1.
%C A369448 Restricted growth sequence transform of the triplet [A003415(n), A327858(n), A359589(n)].
%C A369448 For all i, j: A305800(i) = A305800(j) => a(i) = a(j) => A366297(i) = A366297(j).
%H A369448 Antti Karttunen, <a href="/A369448/b369448.txt">Table of n, a(n) for n = 1..65537</a>
%o A369448 (PARI)
%o A369448 up_to = 65537;
%o A369448 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 A369448 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v.
%o A369448 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A369448 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A369448 A327858(n) = gcd(A003415(n), A276086(n));
%o A369448 v359589 = DirInverseCorrect(vector(up_to,n,A327858(n)-1));
%o A369448 A359589(n) = v359589[n];
%o A369448 Aux369448(n) = [A003415(n), A327858(n), A359589(n)];
%o A369448 v369448 = rgs_transform(vector(up_to, n, Aux369448(n)));
%o A369448 A369448(n) = v369448[n];
%Y A369448 Cf. A003415, A276086, A305800, A327858, A359589, A359595, A366297.
%Y A369448 Cf. also A351236, A369047.
%K A369448 nonn
%O A369448 1,2
%A A369448 _Antti Karttunen_, Jan 26 2024