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%I #20 Jun 07 2018 22:12:01
%S 1,2,2,2,3,2,4,3,2,4,4,3,5,2,4,4,4,6,3,6,5,2,4,4,4,6,4,7,6,3,6,6,5,8,
%T 2,4,4,4,6,4,7,6,4,7,7,6,9,3,6,6,6,10,5,9,8,2,4,4,4,6,4,7,6,4,7,7,6,9,
%U 4,7,7,7,11,6,11,9,3,6,6,6,10,6,11,10,5,9,9,8,12,2,4,4,4,6,4,7,6,4,7,7,6,9,4,7,7,7
%N Restricted growth sequence transform of A278222(A048679(n)).
%C Positions of 2's is given by the positive Fibonacci numbers: 1, 2, 3, 5, 8, 13, 21, ..., that is, A000045(n) from n >= 2 onward.
%C Positions of 3's is given by Lucas numbers larger than 3: 4, 7, 11, 18, ..., that is, A000032(n) from n >= 3 onward.
%C Sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of A048679(n). Compare to the scatter plot of A286622.
%H Antti Karttunen, <a href="/A304101/b304101.txt">Table of n, a(n) for n = 0..65537</a>
%o (PARI)
%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 A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
%o A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
%o A106151(n) = if(n<=1, n, if(n%2, 1+(2*A106151((n-1)/2)), A106151(n>>valuation(n, 2))<<(valuation(n, 2)-1)));
%o A048679(n) = if(!n,n,A106151(2*A003714(n)));
%o A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
%o A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
%o A278222(n) = A046523(A005940(1+n));
%o v304101 = rgs_transform(vector(1+up_to, n, A278222(A048679(n-1))));
%o A304101(n) = v304101[1+n];
%Y Cf. A000032, A000045, A003603, A007895, A035513, A048679, A278222, A304100, A304102, A304103, A304104, A304105.
%Y Cf. also A286622 (compare the scatter-plots).
%K nonn,look
%O 0,2
%A _Antti Karttunen_, May 13 2018
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