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A336395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(A000265(i)) = A278221(A000265(j)), for all i, j >= 1.

%I A336395 #8 Aug 10 2020 09:30:07
%S A336395 1,1,2,1,3,2,4,1,2,3,5,2,6,4,7,1,8,2,9,3,10,5,11,2,3,6,2,4,12,7,13,1,
%T A336395 14,8,15,2,16,9,17,3,18,10,19,5,7,11,20,2,4,3,21,6,22,2,14,4,23,12,24,
%U A336395 7,25,13,10,1,26,14,27,8,28,15,29,2,30,16,7,9,31,17,32,3,2,18,33,10,34,19,35,5,36,7,17,11,37,20,38,2,39,4,14,3,40,21,41,6,42
%N A336395 Lexicographically earliest infinite sequence such that a(i) = a(j) => A278221(A000265(i)) = A278221(A000265(j)), for all i, j >= 1.
%C A336395 Restricted growth sequence transform of the function f(n) = A278221(A000265(n)), the prime signature of the conjugated prime factorization of the odd part of n.
%C A336395 For all i, j:
%C A336395   A324400(i) = A324400(j) => A003602(i) = A003602(j) => a(i) = a(j),
%C A336395   a(i) = a(j) => A005087(i) = A005087(j).
%H A336395 Antti Karttunen, <a href="/A336395/b336395.txt">Table of n, a(n) for n = 1..65537</a>
%H A336395 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%o A336395 (PARI)
%o A336395 up_to = 65537;
%o A336395 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 A336395 A000265(n) = (n>>valuation(n,2));
%o A336395 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
%o A336395 A122111(n) = if(1==n,n,my(f=factor(n), es=Vecrev(f[,2]),is=concat(apply(primepi,Vecrev(f[,1])),[0]),pri=0,m=1); for(i=1, #es, pri += es[i]; m *= prime(pri)^(is[i]-is[1+i])); (m));
%o A336395 A278221(n) = A046523(A122111(n));
%o A336395 v336395 = rgs_transform(vector(up_to, n, A278221(A000265(n))));
%o A336395 A336395(n) = v336395[n];
%Y A336395 Cf. A000265, A003602, A005087, A278221, A286621, A324400.
%Y A336395 Cf. also A336393.
%K A336395 nonn
%O A336395 1,3
%A A336395 _Antti Karttunen_, Aug 10 2020