%I #9 Oct 08 2023 09:02:36
%S 1,2,3,4,5,6,5,4,4,7,5,4,5,7,8,4,5,4,5,4,9,7,5,4,10,7,11,4,5,12,5,4,9,
%T 7,3,4,5,7,9,4,5,13,5,4,13,7,5,4,4,14,8,4,5,15,16,4,8,7,5,4,5,7,17,4,
%U 1,13,5,4,9,6,5,4,5,7,18,4,19,13,5,4,20,7,5,4,3,7,9,4,5,21,19,4,9,7,1,4,5,4,17
%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A366258(i) = A366258(j) for all i, j >= 1, where A366258 is Dirichlet inverse of gcd(n, A366275(n)).
%C Restricted growth sequence transform of A366258.
%H Antti Karttunen, <a href="/A366259/b366259.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</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 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 A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
%o A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
%o A163511(n) = if(!n, 1, my(p=2, t=1); while(n>1, if(!(n%2), (t*=p), p=nextprime(1+p)); n >>= 1); (t*p));
%o A366275(n) = A163511(A057889(n));
%o A366283(n) = gcd(n,A366275(n));
%o v366259 = rgs_transform(DirInverseCorrect(vector(up_to,n,A366283(n))));
%o A366259(n) = v366259[n];
%Y Cf. A366275, A366283, A366258.
%K nonn
%O 1,2
%A _Antti Karttunen_, Oct 07 2023