A286622 Restricted growth sequence computed for filter-sequence A278222, related to 1-runs in the binary representation of n.

1, 2, 2, 3, 2, 4, 3, 5, 2, 4, 4, 6, 3, 6, 5, 7, 2, 4, 4, 6, 4, 8, 6, 9, 3, 6, 6, 10, 5, 9, 7, 11, 2, 4, 4, 6, 4, 8, 6, 9, 4, 8, 8, 12, 6, 12, 9, 13, 3, 6, 6, 10, 6, 12, 10, 14, 5, 9, 9, 14, 7, 13, 11, 15, 2, 4, 4, 6, 4, 8, 6, 9, 4, 8, 8, 12, 6, 12, 9, 13, 4, 8, 8, 12, 8, 16, 12, 17, 6, 12, 12, 18, 9, 17, 13, 19, 3, 6, 6, 10, 6, 12, 10, 14, 6, 12, 12, 18, 10, 18

Offset

0,2

Comments

When filtering sequences (by equivalence class partitioning), this sequence can be used instead of A278222, because for all i, j it holds that: a(i) = a(j) <=> A278222(i) = A278222(j).

For example, for all i, j: a(i) = a(j) => A000120(i) = A000120(j), and for all i, j: a(i) = a(j) => A001316(i) = A001316(j).

The sequence allots a distinct value for each distinct multiset formed from the lengths of 1-runs in the binary representation of n. See the examples. - Antti Karttunen, Jun 04 2017

Links

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to binary expansion of n

Example

For n = 0, there are no 1-runs, thus the multiset is empty [], and it is allotted the number 1, thus a(0) = 1.
For n = 1, in binary also "1", there is one 1-run of length 1, thus the multiset is [1], which has not been encountered before, and a new number is allotted for that, thus a(1) = 2.
For n = 2, in binary "10", there is one 1-run of length 1, thus the multiset is [1], which was already encountered at n=1, thus a(2) = a(1) = 2.
For n = 3, in binary "11", there is one 1-run of length 2, thus the multiset is [2], which has not been encountered before, and a new number is allotted for that, thus a(3) = 3.
For n = 4, in binary "100", there is one 1-run of length 1, thus the multiset is [1], which was already encountered at n=1 for the first time, thus a(4) = a(1) = 2.
For n = 5, in binary "101", there are two 1-runs, both of length 1, thus the multiset is [1,1], which has not been encountered before, and a new number is allotted for that, thus a(5) = 4.

Prog

(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));
v286622 = rgs_transform(vector(1+65537, n, A278222(n-1)));
A286622(n) = v286622[1+n];

Crossrefs

Cf. A278222, A000120, A001316.

Cf. A286552 (ordinal transform).

Cf. also A101296, A286581, A286589, A286597, A286599, A286600, A286602, A286603, A286605, A286610, A286619, A286621, A286626, A286378, A304101 for similarly constructed or related sequences.

Cf. also A305793, A305795.

Sequence in context: A364236 A356276 A286532 * A286589 A246029 A245564

Adjacent sequences: A286619 A286620 A286621 * A286623 A286624 A286625

Keyword

nonn,look

Author

Antti Karttunen, May 11 2017

Extensions

Example section added by Antti Karttunen, Jun 04 2017

Status

approved