A175021 A positive integer k is included if k is not the smallest positive integer with its particular multiset of run-lengths (of either 0 or 1 considered together) in its binary representation.

6, 11, 13, 14, 20, 22, 23, 25, 26, 27, 28, 29, 30, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 70, 72, 75, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109

Offset

1,1

Comments

A175020 contains those positive integers not in this sequence.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

9 in binary is 1001. The run lengths form the multiset (1,2,1). Since no positive integer < 9 has this same multiset of run lengths, then 9 is not in this sequence.
On the other hand, 23 in binary is 10111. The run-lengths are (1,1,3). But 17 (which is < 23) in binary is 10001, which has the run-lengths of (1,3,1). Since the multisets (1,1,3) and (1,3,1) are identical, 23 is in this sequence.

Mathematica

Block[{nn = 109, s}, s = Array[Sort@ Map[Length, Split@ IntegerDigits[#, 2]] &, nn]; Complement[Range[nn], Values[PositionIndex@ s][[All, 1]] ]] (* Michael De Vlieger, Sep 03 2017 *)

Prog

(PARI) {A175021_first(n)=my(k, S=[]); vector(n, R, while(!setsearch(S, R=vecsort(A101211_row(k++))), S=setunion(S, [R])); k)} \\ M. F. Hasler, Mar 11 2025
(Python)
from itertools import count, groupby
def A175021(n=None, stop=3**33): # return a(n) or generator of sequence if no n given
    if not n: return(A175021(n)for n in range(1, stop))
    if not hasattr(A := A175021, 'terms'): A.terms=[0]; A.rls=set(); A.signature=\
        lambda n: tuple(sorted(len(list(g)) for _, g in groupby(bin(n)[2:])))
    while n>=len(A.terms):
       A.terms.append(next(k for k in count(A.terms[-1]+1)
           if(s:=A.signature(k)) in A.rls or A.rls.add(s)))
    return A.terms[n] # M. F. Hasler, Mar 11 2025

Crossrefs

Cf. A175020.

Sequence in context: A219553 A015870 A004471 * A315367 A046953 A121765

Adjacent sequences: A175018 A175019 A175020 * A175022 A175023 A175024

Keyword

nonn,base,changed

Author

Leroy Quet, Nov 03 2009

Extensions

Extended by Ray Chandler, Mar 11 2010

Status

approved