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A352898 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A352892(n)], except f(n) = -n when <= 2.

%I A352898 #12 Apr 09 2022 13:24:42
%S A352898 1,2,3,4,3,5,3,6,7,8,3,9,3,10,11,12,3,13,3,14,15,16,3,17,7,18,19,20,3,
%T A352898 21,3,22,23,24,11,25,3,26,27,28,3,29,3,30,31,32,3,33,7,34,35,36,3,37,
%U A352898 15,38,39,40,3,41,3,42,34,43,23,44,3,45,46,47,3,48,3,49,50,51,11,52,3,53,54,55,3,56,27,57,58,59,3,60,15
%N A352898 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A046523(n), A352892(n)], except f(n) = -n when <= 2.
%C A352898 For all i, j: A305801(i) = A305801(j) => a(i) = a(j) => A352897(i) = A352897(j).
%H A352898 Antti Karttunen, <a href="/A352898/b352898.txt">Table of n, a(n) for n = 1..65537</a>
%H A352898 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%H A352898 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%o A352898 (PARI)
%o A352898 up_to = 65537;
%o A352898 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 A352898 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
%o A352898 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A352898 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A352898 A156552(n) = { my(f = factor(n), p, p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
%o A352898 A329603(n) = A005940(2+(3*A156552(n)));
%o A352898 A341515(n) = if(n%2, A064989(n), A329603(n));
%o A352898 A348717(n) = { my(f=factor(n)); if(#f~>0, my(pi1=primepi(f[1, 1])); for(k=1, #f~, f[k, 1] = prime(primepi(f[k, 1])-pi1+1))); factorback(f); }; \\ From A348717
%o A352898 A352892(n) = A348717(A341515(n));
%o A352898 Aux352898(n) = if(n<=2,-n,[A046523(n),A352892(n)]);
%o A352898 v352898 = rgs_transform(vector(up_to, n, Aux352898(n)));
%o A352898 A352898(n) = v352898[n];
%Y A352898 Cf. A046523, A101296, A305801, A305897, A352892, A352897, A352899.
%K A352898 nonn
%O A352898 1,2
%A A352898 _Antti Karttunen_, Apr 08 2022