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Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A003557(A332824(n)) for all other numbers, except f(1) = 0.
3

%I #8 Feb 25 2020 18:34:22

%S 1,2,3,2,3,4,3,2,5,6,3,7,3,8,3,2,3,9,3,6,10,11,3,12,13,14,15,16,3,17,

%T 3,2,5,18,13,19,3,20,21,22,3,23,3,24,25,26,3,27,5,28,3,29,3,30,21,31,

%U 5,32,3,33,3,34,35,2,13,36,3,18,10,37,3,38,3,39,40,41,25,42,3,43,44,45,3,46,3,47,48,49,3,50,40,51,52,53,3,54,3,55,56,57,3,58,3,59,60

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = A003557(A332824(n)) for all other numbers, except f(1) = 0.

%C For all i, j:

%C A305801(i) = A305801(j) => a(i) = a(j).

%H Antti Karttunen, <a href="/A332827/b332827.txt">Table of n, a(n) for n = 1..65537</a>

%o (PARI) up_to = 65537;

%o 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 A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565

%o A332824(n) = { my(m=1); fordiv(n,d,m *= A019565(eulerphi(d))); (m); };

%o A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); };

%o Aux332827(n) = if(1==n,0,A003557(A332824(n)));

%o v332827 = rgs_transform(vector(up_to,n,Aux332827(n)));

%o A332827(n) = v332827[n];

%Y Cf. A000010, A003557, A019565, A305801, A332824, A332826.

%Y Cf. also A324538.

%K nonn

%O 1,2

%A _Antti Karttunen_, Feb 25 2020