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A286619 Restricted growth sequence computed for filter-sequence A278219, related to run-lengths in the binary representation of n.

%I A286619 #15 May 14 2017 00:04:09
%S A286619 1,2,3,2,3,4,5,2,3,6,7,4,5,6,5,2,3,6,8,6,7,9,10,4,5,11,10,6,5,6,5,2,3,
%T A286619 6,8,6,8,12,13,6,7,14,15,9,10,12,10,4,5,11,13,11,10,14,13,6,5,11,10,6,
%U A286619 5,6,5,2,3,6,8,6,8,12,13,6,8,16,17,12,13,16,13,6,7,14,17,14,15,18,19,9,10,20,21,12,10,12,10,4,5,11,13,11,13,20,22,11,10
%N A286619 Restricted growth sequence computed for filter-sequence A278219, related to run-lengths in the binary representation of n.
%C A286619 When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A278219, because for all i, j it holds that: a(i) = a(j) <=> A278219(i) = A278219(j).
%C A286619 For example, for all i, j: a(i) = a(j) => A005811(i) = A005811(j). (The same is true for A073334, as it is a sequence computed from A005811).
%H A286619 Antti Karttunen, <a href="/A286619/b286619.txt">Table of n, a(n) for n = 0..65537</a>
%H A286619 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%t A286619 f[n_, i_, x_] := Which[n == 0, x, EvenQ@ n, f[n/2, i + 1, x], True, f[(n - 1)/2, i, x Prime@ i]]; g[n_] := If[n == 1, 1, Times @@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]]; With[{nn = 99}, Function[s, Table[Position[Keys@ s, k_ /; MemberQ[k, n]][[1, 1]], {n, nn}]]@ Map[#1 -> #2 & @@ # &, Transpose@ {Values@ #, Keys@ #}] &@ PositionIndex@ Table[g@ f[BitXor[n, Floor[n/2]], 1, 1], {n, 0, nn}]] (* _Michael De Vlieger_, May 12 2017, Version 10 *)
%o A286619 (PARI)
%o A286619 rgs_transform(invec) = { my(occurrences = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(occurrences,invec[i]), my(pp = mapget(occurrences, invec[i])); outvec[i] = outvec[pp] , mapput(occurrences,invec[i],i); outvec[i] = u; u++ )); outvec; };
%o A286619 write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
%o A286619 A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t }; \\ Modified from code of _M. F. Hasler_
%o A286619 A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from _Charles R Greathouse IV_, Aug 17 2011
%o A286619 A278222(n) = A046523(A005940(1+n));
%o A286619 A003188(n) = bitxor(n, n>>1);
%o A286619 A278219(n) = A278222(A003188(n));
%o A286619 write_to_bfile(0,rgs_transform(vector(65538,n,A278219(n-1))),"b286619.txt");
%Y A286619 Cf. A278219, A005811, A073334.
%Y A286619 Cf. also A101296, A286603, A286605, A286610, A286621, A286622, A286626, A286378 for similarly constructed sequences.
%K A286619 nonn,look
%O A286619 0,2
%A A286619 _Antti Karttunen_, May 11 2017