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A366384 Lexicographically earliest infinite sequence such that a(i) = a(j) => A355828(i) = A355828(j) for all i, j >= 1, where A355828 is Dirichlet inverse of A342671, the greatest common divisor of sigma(n) and A003961(n).

%I A366384 #9 Oct 12 2023 20:47:59
%S A366384 1,2,3,4,3,5,3,6,7,5,3,8,3,5,1,9,3,7,3,10,1,5,3,11,7,5,12,8,3,2,3,13,
%T A366384 1,5,1,7,3,5,1,14,3,2,3,15,7,5,3,7,7,7,1,8,3,11,1,16,2,5,3,17,3,5,7,
%U A366384 18,19,2,3,20,1,2,3,11,3,5,7,8,1,2,3,21,4,5,3,4,1,5,2,22,3,7,1,15,1,5,1,23,3,7,24
%N A366384 Lexicographically earliest infinite sequence such that a(i) = a(j) => A355828(i) = A355828(j) for all i, j >= 1, where A355828 is Dirichlet inverse of A342671, the greatest common divisor of sigma(n) and A003961(n).
%H A366384 Antti Karttunen, <a href="/A366384/b366384.txt">Table of n, a(n) for n = 1..65537</a>
%o A366384 (PARI)
%o A366384 up_to = 65537;
%o A366384 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 A366384 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 A366384 A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
%o A366384 A342671(n) = gcd(sigma(n), A003961(n));
%o A366384 v366384 = rgs_transform(DirInverseCorrect(vector(up_to,n,A342671(n))));
%o A366384 A366384(n) = v366384[n];
%Y A366384 Cf. A000203, A003961, A342671, A355828.
%K A366384 nonn
%O A366384 1,2
%A A366384 _Antti Karttunen_, Oct 12 2023