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A365467 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336467(i) = A336467(j) and A336467(A163511(i)) = A336467(A163511(j)) for all i, j >= 1, where A336467 is fully multiplicative with a(2) = 1 and a(p) = oddpart(p+1) for odd primes p.

%I A365467 #8 Sep 04 2023 18:23:48
%S A365467 1,1,1,1,2,1,3,1,1,2,4,1,5,3,2,1,6,1,7,2,8,4,2,1,9,5,3,3,10,2,3,1,2,6,
%T A365467 11,1,12,7,13,2,14,8,15,4,2,2,4,1,3,9,16,5,17,3,9,3,18,10,19,2,20,3,
%U A365467 21,1,22,2,23,6,11,11,6,1,24,12,9,7,2,13,7,2,8,14,14,8,17,15,19,4,25,2,26,2,8,4,27,1,28,3
%N A365467 Lexicographically earliest infinite sequence such that a(i) = a(j) => A336467(i) = A336467(j) and A336467(A163511(i)) = A336467(A163511(j)) for all i, j >= 1, where A336467 is fully multiplicative with a(2) = 1 and a(p) = oddpart(p+1) for odd primes p.
%C A365467 Restricted growth sequence transform of the ordered pair [A336467(n), A365427(n)].
%C A365467 For all i, j: A003602(i) = A003602(j) => a(i) = a(j).
%H A365467 Antti Karttunen, <a href="/A365467/b365467.txt">Table of n, a(n) for n = 1..65537</a>
%o A365467 (PARI)
%o A365467 up_to = 65537;
%o A365467 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 A365467 A000265(n) = (n>>valuation(n,2));
%o A365467 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 A365467 A336467(n) = { my(f=factor(n)); prod(k=1,#f~,if(2==f[k,1],1,(A000265(f[k,1]+1))^f[k,2])); };
%o A365467 A365467aux(n) = [A336467(n), A336467(A163511(n))];
%o A365467 v365467 = rgs_transform(vector(up_to,n,A365467aux(n)));
%o A365467 A365467(n) = v365467[n];
%Y A365467 Cf. A000265, A003602, A163511, A336467, A365427.
%Y A365467 Cf. also A365466.
%K A365467 nonn
%O A365467 1,5
%A A365467 _Antti Karttunen_, Sep 04 2023