A261461 -id:A261461 - cp's OEIS frontend

A261794 a(n) is the smallest nonzero number that is not a substring of n in decimal representation.

1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1

Offset: 0

Reinhard Zumkeller, Sep 01 2015

A261795(n) = a(A261793(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007089, A031298, A261795, A261793, A261461, A261787.

import Data.List (isInfixOf)
a261794 x = f $ tail a031298_tabf where
   f (cs:css) = if isInfixOf cs (a031298_row x)
                   then f css else foldr (\d v -> 10 * v + d) 0 cs

Name corrected by Álvar Ibeas, Sep 08 2020

A261923 Number of steps to reach 0, starting at n, and iteration the map x -> A261922(x).

Original entry on oeis.org

0, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 3, 3, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 3, 2, 3, 3, 2, 2, 2, 4, 2, 2

Offset: 0

N. J. A. Sloane, Sep 17 2015

nonn

			13 -> 4 -> 3 -> 0, which takes 3 steps to reach 0, so a(13)=3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A261922, A261461, A262279.

a261923 n = fst $ until ((== 0) . snd)
                        (\(step, x) -> (step + 1, a261922 x)) (0, n)
-- Reinhard Zumkeller, Sep 17 2015

a(n) = if (n==0, 0, my(k=1, x=A261922(n)); while (x, x=A261922(x); k++); k); \\ Michel Marcus, Sep 20 2023

def f(n): b=bin(n)[2:]; return next(k for k in range(2**len(b)) if bin(k)[2:] not in b)
def a(n): return 0 if n == 0 else 1 + a(f(n))
print([a(n) for n in range(99)]) # Michael S. Branicky, Sep 21 2023

a(A262279(n)) = n. - Reinhard Zumkeller, Sep 17 2015

A262281 a(n) = smallest nonnegative number, not a power of 2, that is not a substring of n in its binary representation.

Original entry on oeis.org

3, 0, 3, 0, 3, 3, 5, 0, 3, 3, 3, 6, 5, 7, 5, 0, 3, 3, 3, 5, 3, 3, 7, 6, 5, 5, 7, 7, 5, 9, 5, 0, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 7, 7, 9, 6, 5, 5, 5, 5, 7, 7, 7, 9, 5, 5, 9, 9, 5, 9, 5, 0, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 5, 7, 5, 5, 3, 3, 3, 6, 3, 3, 7

Offset: 0

N. J. A. Sloane, Sep 17 2015

Similar to A261922, but if the smallest missing number is a power of 2, ignore it and look at the next-smallest missing number.

This is like applying A261922 not to n itself but to n plus a very large power of 2. Suggested by considering A261416.

			For n = 13 = 1101_2, we can see 0, 11 (3), 101 (5), 110 (6), but not 111 (7), so a(13)=7.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..10000

Cf. A261416, A261461, A261922.

See A262289 for the "positive" version.

a(23)-a(86) from Hiroaki Yamanouchi, Sep 20 2015

A262289 a(n) = smallest positive number, not a power of 2, that is not a substring of n in its binary representation.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 5, 7, 5, 5, 3, 3, 3, 5, 3, 3, 7, 6, 5, 5, 7, 7, 5, 9, 5, 5, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 7, 7, 9, 6, 5, 5, 5, 5, 7, 7, 7, 9, 5, 5, 9, 9, 5, 9, 5, 5, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 5, 7, 5, 5, 3, 3, 3, 6, 3, 3, 7

Offset: 0

N. J. A. Sloane, Sep 19 2015

Similar to A261461, but if the smallest missing number is a power of 2, ignore it and look at the next-smallest missing number.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..10000
David Consiglio, Jr., Python Program

Cf. A261416, A261461, A261922.

See A262281 for the "nonnegative" version.

fQ[m_, n_] := Block[{g}, g[x_] := ToString@FromDigits@IntegerDigits[x, 2]; StringContainsQ[g@ n, g@ m]]; Table[k = 3; While[Or[fQ[k, n] && k < 2 n, IntegerQ@ Log[2, k]], k++]; k, {n, 0, 86}] (* Michael De Vlieger, Sep 21 2015 *)

a(23)-a(86) from Hiroaki Yamanouchi, Sep 20 2015

A301984 a(n) is the greatest positive number k such that the binary digits of any number in the interval 1..k appear in order but not necessarily as consecutive digits in the binary representation of n.

Original entry on oeis.org

1, 2, 1, 2, 3, 3, 1, 2, 5, 6, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 7, 7, 3, 4, 7, 7, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 11, 11, 5, 6, 13, 14, 7, 8, 7, 7, 3, 4, 9, 10, 7, 8, 7, 7, 3, 4, 7, 7, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 11, 11, 5, 6, 13, 14, 11, 12, 11, 11, 5, 6, 13, 14

Offset: 1

Rémy Sigrist, Mar 30 2018

Equivalently, a(n) is the greatest positive number k such that A301983(n, k) = k.

Apparently, the k-th record value is A089633(k), and the first term with this value has index A048678(A089633(k)).

			The 13th row of A301983 is: 1, 2, 3, 5, 6, 7, 13; all numbers in the range 1..3 appear in this row, but the number 4 is missing; hence a(13) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A048678, A089633, A261461, A301977, A301983.

a(n) = my (b=binary(n), s=Set(1)); for (i=2, #b, s = setunion(s, Set(apply(v -> 2*v+b[i], s)))); for (u=1, oo, if (!setsearch(s,u), return (u-1)))

a(n) <= A301977(n).
a(2*n) >= a(n).
a(2*n + 1) >= a(n) (with strict inequality if a(n) is even).
a(n) = 1 iff n is positive and belongs to A000225.

cp's OEIS Frontend

A261794 a(n) is the smallest nonzero number that is not a substring of n in decimal representation.

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A261923 Number of steps to reach 0, starting at n, and iteration the map x -> A261922(x).

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A262281 a(n) = smallest nonnegative number, not a power of 2, that is not a substring of n in its binary representation.

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A262289 a(n) = smallest positive number, not a power of 2, that is not a substring of n in its binary representation.

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Author

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Mathematica

Extensions

A301984 a(n) is the greatest positive number k such that the binary digits of any number in the interval 1..k appear in order but not necessarily as consecutive digits in the binary representation of n.

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