cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

Antti Karttunen, May 11 2017

Keywords

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

Examples

			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.

Links

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.

Programs

  • 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];

Extensions

Example section added by Antti Karttunen, Jun 04 2017

A263017 n is the a(n)-th positive integer having its binary weight.

Original entry on oeis.org

1, 2, 1, 3, 2, 3, 1, 4, 4, 5, 2, 6, 3, 4, 1, 5, 7, 8, 5, 9, 6, 7, 2, 10, 8, 9, 3, 10, 4, 5, 1, 6, 11, 12, 11, 13, 12, 13, 6, 14, 14, 15, 7, 16, 8, 9, 2, 15, 17, 18, 10, 19, 11, 12, 3, 20, 13, 14, 4, 15, 5, 6, 1, 7, 16, 17, 21, 18, 22, 23, 16, 19, 24, 25, 17
Offset: 1

Views

Author

Paul Tek, Oct 07 2015

Keywords

Comments

Binary weight is given by A000120.
a(2^k) = k+1 for any k>=0.
a(2^k-1) = 1 for any k>0.
a(A057168(k)) = a(k)+1 for any k>0.
a(A036563(k+1)) = k for any k>0.
Ordinal transform of A000120. - Alois P. Heinz, Dec 23 2018

Examples

			The numbers with binary weight 3 are: 7, 11, 13, 14, 19, ...
Hence: a(7)=1, a(11)=2, a(13)=3, a(14)=4, a(19)=5, ...
And more generally: a(A014311(k))=k for any k>0.

Links

Crossrefs

One more than A068076.
Cf. A000120, A036563, A057168, A254524, A286478, A286552, A286554, A286558.
Cf. A263109, A263110.

Programs

  • Haskell
    import Data.IntMap (empty, findWithDefault, insert)
a263017 n = a263017_list !! (n-1)
a263017_list = f 1 empty where
   f x m = y : f (x + 1) (insert h (y + 1) m) where
           y = findWithDefault 1 h m
           h = a000120 x
-- Reinhard Zumkeller, Oct 09 2015
  • Maple
    a:= proc() option remember; local a, b, t; b, a:=
      proc() 0 end, proc(n) option remember; a(n-1);
        t:= add(i, i=convert(n, base, 2)); b(t):= b(t)+1
      end; a(0):=0; a
    end():
seq(a(n), n=1..120);  # Alois P. Heinz, Dec 23 2018
  • Perl
    # See Links section.
  • Python
    from math import comb
def A263017(n):
    c, k = 1, 0
    for i,j in enumerate(bin(n)[-1:1:-1]):
        if j == '1':
            k += 1
            c += comb(i,k)
    return c # Chai Wah Wu, Mar 01 2023

Formula

a(n) = 1 + A068076(n). - Antti Karttunen, May 22 2017

A286554 Ordinal transform of A286619, or equally, of A278219.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 2, 3, 4, 4, 3, 1, 4, 2, 1, 1, 3, 4, 1, 2, 5, 5, 6, 6, 5, 5, 7, 2, 8, 3, 1, 1, 9, 3, 1, 1, 2, 3, 2, 4, 4, 7, 2, 2, 3, 5, 2, 3, 10, 8, 4, 6, 11, 9, 12, 10, 6, 6, 13, 4, 14, 5, 3, 4, 15, 6, 1, 1, 4, 5, 2, 6, 16, 4, 3, 2, 4, 2, 1, 1, 3, 7, 1, 1, 5, 8, 6, 9, 5, 11, 5, 7, 6, 8, 2, 1, 7, 10, 3, 2, 5, 9, 3, 10, 17, 12, 8, 11, 9
Offset: 0

Views

Author

Antti Karttunen, May 21 2017

Keywords

Links

Crossrefs

Cf. A278219, A286619, A286552, A286558.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.