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

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, 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, 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

Offset: 0

Author

Antti Karttunen, May 11 2017

Keywords

• nonn
• look

Comments

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).

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).

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A278219, A005811, A073334.

Cf. also A101296, A286603, A286605, A286610, A286621, A286622, A286626, A286378 for similarly constructed sequences.

Programs

Mathematica

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 *)

PARI

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; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
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
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
A278222(n) = A046523(A005940(1+n));
A003188(n) = bitxor(n, n>>1);
A278219(n) = A278222(A003188(n));
write_to_bfile(0,rgs_transform(vector(65538,n,A278219(n-1))),"b286619.txt");