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A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).

%I A323372 #7 Jan 13 2019 19:29:33
%S A323372 1,2,3,4,5,6,7,8,9,10,11,12,13,14,14,15,16,17,18,19,20,21,22,23,24,25,
%T A323372 26,27,28,29,30,31,32,33,32,34,35,36,37,38,39,40,41,42,43,44,45,46,47,
%U A323372 48,44,49,50,51,44,52,53,54,55,56,57,58,59,60,61,62,63,64,58,62,29,65,66,67,68,69,58,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,79
%N A323372 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => A003557(i) = A003557(j) and A323363(i) = A323363(j).
%C A323372 Restricted growth sequence transform of ordered pair [A003557(n), A323363(n)].
%C A323372 For all i, j:
%C A323372   a(i) = a(j) => A291751(i) = A291751(j),
%C A323372   a(i) = a(j) => A323364(i) = A323364(j).
%H A323372 Antti Karttunen, <a href="/A323372/b323372.txt">Table of n, a(n) for n = 1..65537</a>
%o A323372 (PARI) up_to = 65537;
%o A323372 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 A323372 DirInverse(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = -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 A323372 A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ From A001615
%o A323372 v323363 = DirInverse(vector(up_to,n,A001615(n)));
%o A323372 A323363(n) = v323363[n];
%o A323372 A003557(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };
%o A323372 v323372 = rgs_transform(vector(up_to, n, [A003557(n), A323363(n)]));
%o A323372 A323372(n) = v323372[n];
%Y A323372 Cf. A001615, A003557, A291750, A291751, A323363, A323364.
%K A323372 nonn
%O A323372 1,2
%A A323372 _Antti Karttunen_, Jan 13 2019