%I #12 Jan 29 2022 22:32:05

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

%T 4,9,4,14,2,4,9,15,2,16,2,7,7,17,2,18,3,19,9,7,2,10,6,10,9,20,2,21,2,

%U 9,7,22,4,12,2,11,4,16,2,23,2,9,7,11,4,12,2,18,8,9,2,24,9,25,17,26,2,24,6,11,20,27,9

%N Lexicographically earliest infinite sequence such that a(i) = a(j) => A046523(i) = A046523(j) and A350063(i) = A350063(j), for all i, j >= 1.

%C Restricted growth sequence transform of the ordered pair [A046523(n), A350063(n)].

%C For all i, j >= 1: A305897(i) = A305897(j) => a(i) = a(j).

%H Antti Karttunen, <a href="/A350068/b350068.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 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 A350063(n) = if(1==n,0,A046523(A000265(A156552(n))));

%o Aux350068(n) = [A046523(n),A350063(n)];

%o v350068 = rgs_transform(vector(up_to, n, Aux350068(n)));

%o A350068(n) = v350068[n];

%Y Cf. A000265, A046523, A156552, A322993, A350063.

%Y Cf. A000040 (positions of 2's), A001248 (of 3's).

%Y Cf. also A101296, A300226, A305897, A350065.

%K nonn

%O 1,2

%A _Antti Karttunen_, Jan 29 2022