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A353520 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A053669(i) = A053669(j), for all i, j >= 1.

%I A353520 #8 Apr 25 2022 16:57:09
%S A353520 1,2,3,4,3,5,3,6,7,8,3,9,3,10,11,12,3,13,3,14,15,16,3,17,18,19,20,21,
%T A353520 3,22,3,23,24,25,26,27,3,28,29,30,3,31,3,32,33,34,3,35,36,37,38,39,3,
%U A353520 40,29,41,42,43,3,44,3,45,46,47,48,49,3,50,51,52,3,53,3,54,55,56,48,57,3,58,59,60,3,61,42,62,63,64,3,65,38
%N A353520 Lexicographically earliest infinite sequence such that a(i) = a(j) => A003415(i) = A003415(j), A003557(i) = A003557(j) and A053669(i) = A053669(j), for all i, j >= 1.
%C A353520 Restricted growth sequence transform of the triplet [A003415(n), A003557(n), A053669(n)].
%C A353520 For all i, j:
%C A353520   A305801(i) = A305801(j) => a(i) = a(j),
%C A353520   a(i) = a(j) => A007814(i) = A007814(j),
%C A353520   a(i) = a(j) => A344025(i) = A344025(j).
%H A353520 Antti Karttunen, <a href="/A353520/b353520.txt">Table of n, a(n) for n = 1..65537</a>
%o A353520 (PARI)
%o A353520 up_to = 65537;
%o A353520 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 A353520 A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A353520 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A353520 A053669(n) = forprime(p=2, , if(n%p, return(p))); \\ From A053669
%o A353520 Aux353520(n) = [A003415(n), A003557(n), A053669(n)];
%o A353520 v353520 = rgs_transform(vector(up_to,n,Aux353520(n)));
%o A353520 A353520(n) = v353520[n];
%Y A353520 Cf. A003415, A003557, A007814, A053669.
%Y A353520 Cf. also A305801, A328470, A344025, A351260.
%K A353520 nonn
%O A353520 1,2
%A A353520 _Antti Karttunen_, Apr 25 2022