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%I #12 Jan 29 2022 22:31:59
%S 1,2,2,3,2,3,2,4,3,5,2,6,2,3,3,7,2,4,2,8,5,9,2,10,3,11,4,11,2,11,2,12,
%T 3,13,3,14,2,3,9,11,2,8,2,15,6,16,2,17,3,5,11,18,2,19,5,20,13,21,2,22,
%U 2,23,8,24,3,25,2,26,3,6,2,9,2,27,4,28,3,26,2,29,7,30,2,31,9,32,16,33,2,31,5,34,21,35
%N Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = [A342666(n), A350063(n)] for n > 1, with f(1) = 1.
%C Restricted growth sequence transform of the ordered pair [A342666(n), A350063(n)], when assuming that A342666(1) = 0.
%C Restricted growth sequence transform of the function f(1) = 0, f(n) = A336470(A156552(n)) for n > 1.
%C For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j) => A350065(i) = A350065(j).
%C For all i, j >= 2: a(i) = a(j) => A342651(i) = A342651(j).
%H Antti Karttunen, <a href="/A350067/b350067.txt">Table of n, a(n) for n = 1..10000</a> (based on Hans Havermann's factorization of A156552)
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%o (PARI)
%o up_to = 3003;
%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 A000265(n) = (n>>valuation(n,2));
%o A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
%o A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
%o 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 A336466(n) = { my(f=factor(n)); prod(k=1, #f~, A000265(f[k, 1]-1)^f[k, 2]); };
%o Aux350067(n) = if(1==n,1,my(u=A000265(A156552(n))); [A046523(u),A336466(u)]);
%o v350067 = rgs_transform(vector(up_to, n, Aux350067(n)));
%o A350067(n) = v350067[n];
%Y Cf. A156552, A322993, A336466, A342666, A350063, A336470.
%Y Cf. also A305897, A350065, A342651.
%K nonn
%O 1,2
%A _Antti Karttunen_, Jan 29 2022
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