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A350065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350063(i) = A350063(j), for all i, j >= 1.

%I A350065 #13 Jan 29 2022 22:32:14
%S A350065 1,2,2,3,2,3,2,3,3,4,2,3,2,3,3,5,2,3,2,3,4,5,2,3,3,5,3,5,2,5,2,3,3,5,
%T A350065 3,6,2,3,5,5,2,3,2,3,3,7,2,3,3,4,5,3,2,3,4,3,5,8,2,3,2,5,3,8,3,5,2,5,
%U A350065 3,3,2,5,2,5,3,5,3,5,2,3,5,5,2,8,5,9,7,7,2,8,4,5,8,10,5,5,2,4,5,5,2,8,2,3,5
%N A350065 Lexicographically earliest infinite sequence such that a(i) = a(j) => A350063(i) = A350063(j), for all i, j >= 1.
%C A350065 Restricted growth sequence transform of A350063.
%C A350065 For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A324117(i) = A324117(j).
%C A350065 For all i, j >= 2: a(i) = a(j) => A342656(i) = A342656(j).
%H A350065 Antti Karttunen, <a href="/A350065/b350065.txt">Table of n, a(n) for n = 1..10000</a> (based on Hans Havermann's factorization of A156552)
%H A350065 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%o A350065 (PARI)
%o A350065 up_to = 3000;
%o A350065 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 A350065 A000265(n) = (n>>valuation(n,2));
%o A350065 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
%o A350065 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 A350065 A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));
%o A350065 v350065 = rgs_transform(vector(up_to, n, A350063(n)));
%o A350065 A350065(n) = v350065[n];
%o A350065 (PARI)
%o A350065 \\ Version using the factorization file available at https://oeis.org/A156552/a156552.txt
%o A350065 v156552sigs = readvec("a156552.txt");
%o A350065 up_to = #v156552sigs;
%o A350065 A350063(n) = if(n<=2,n-1,my(es=v156552sigs[n][2]); if(n%2, es = vector(#es-1,i,es[1+i])); my(f=vecsort(es, , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]));
%o A350065 v350065 = rgs_transform(vector(up_to, n, A350063(n)));
%o A350065 A350065(n) = v350065[n]; \\ _Antti Karttunen_, Jan 29 2022
%Y A350065 Cf. A000265, A046523, A156552, A322993, A350063, A350064, A350067, A350068.
%Y A350065 Cf. also A305897, A324117, A342656.
%K A350065 nonn
%O A350065 1,2
%A A350065 _Antti Karttunen_, Jan 29 2022