A261461 a(n) is the smallest nonzero number that is not a substring of n in its binary representation.

1, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 4, 5, 4, 4, 2, 3, 3, 3, 5, 3, 3, 4, 4, 5, 5, 4, 4, 5, 4, 4, 2, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 4, 7, 4, 4, 4, 5, 5, 5, 5, 7, 4, 4, 4, 5, 5, 4, 4, 5, 4, 4, 2, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 5, 7, 5, 5, 3, 3, 3, 6, 3, 3

Offset: 0

Author

Reinhard Zumkeller, Aug 30 2015

Keywords

• nonn
• base

Comments

A261018(n) = a(A260273(n)).

Is a(n) = A091460(n) for n>1? - R. J. Mathar, Sep 02 2015. The lowest counterexample occurs at a(121) = 5 < 6 = A091460(121). - Álvar Ibeas, Sep 08 2020

a(A062289(n))=A261922(A062289(n)); a(A126646(n))!=A261922(A126646(n)). - Reinhard Zumkeller, Sep 17 2015

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• David Consiglio, Jr., Python Program

Crossrefs

Cf. A007088, A030308, A260273, A261018; record values and where they occur: A261466, A261467.

See A261922 for a variant.

Cf. A062289, A144016, A261922, A126646.

Programs

Haskell

import Data.List (isInfixOf)
a261461 x = f $ tail a030308_tabf where
   f (cs:css) = if isInfixOf cs (a030308_row x)
                   then f css else foldr (\d v -> 2 * v + d) 0 cs

Mathematica

fQ[m_, n_] := Block[{g}, g[x_] := ToString@ FromDigits@ IntegerDigits[x, 2]; StringContainsQ[g@ n, g@ m]]; Table[k = 1; While[fQ[k, n] && k < n, k++]; k, {n, 85}] (* Michael De Vlieger, Sep 21 2015 *)

Python

from itertools import count
def a(n):
    b, k = bin(n)[2:], 1
    return next(k for k in count(1) if bin(k)[2:] not in b)
print([a(n) for n in range(86)]) # Michael S. Branicky, Feb 26 2023

Formula

a(n) = A144016(n) + 1 for any n > 0. - Rémy Sigrist, Mar 10 2018