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A366296 Lexicographically earliest infinite sequence such that a(i) = a(j) => A346242(i) = A346242(j) for all i, j >= 1, where A346242 is Dirichlet inverse of gcd(n, A276086(n)).

%I A366296 #10 Oct 07 2023 21:39:54
%S A366296 1,2,3,4,2,5,2,4,6,3,2,7,2,1,8,4,2,9,2,10,11,1,2,4,12,1,13,4,2,14,2,4,
%T A366296 11,1,15,16,2,1,11,4,2,17,2,4,18,1,2,4,19,20,11,4,2,21,3,19,11,1,2,22,
%U A366296 2,1,12,4,1,15,2,4,11,23,2,24,2,1,25,4,15,15,2,4,26,1,2,27,3,1,11,4,2,28,15,4,11,1,1,4
%N A366296 Lexicographically earliest infinite sequence such that a(i) = a(j) => A346242(i) = A346242(j) for all i, j >= 1, where A346242 is Dirichlet inverse of gcd(n, A276086(n)).
%C A366296 Restricted growth sequence transform of A346242.
%C A366296 For all i, j: A305900(i) = A305900(j) => a(i) = a(j) => A008966(i) = A008966(j).
%H A366296 Antti Karttunen, <a href="/A366296/b366296.txt">Table of n, a(n) for n = 1..65537</a>
%o A366296 (PARI)
%o A366296 up_to = 65537;
%o A366296 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 A366296 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 A366296 A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
%o A366296 v366296 = rgs_transform(DirInverseCorrect(vector(up_to,n,A324198(n))));
%o A366296 A366296(n) = v366296[n];
%Y A366296 Cf. A008966, A276086, A305900, A324198, A346242.
%Y A366296 Cf. also A366297.
%K A366296 nonn
%O A366296 1,2
%A A366296 _Antti Karttunen_, Oct 07 2023