A261416 -id:A261416 - cp's OEIS frontend

A261926 First differences of A261416.

3, 3, 3, 6, 3, 3, 6, 9, 5, 6, 5, 7, 9, 5, 6, 3, 3, 6, 9, 5, 7, 5, 9, 3, 5, 3, 6, 3, 3, 6, 10, 6, 7, 7, 9, 5, 5, 5, 7, 7, 5, 5, 9, 9, 9, 5, 6, 3, 3, 6, 9, 5, 6, 5, 7, 10, 6, 7, 3, 6, 9, 7, 7, 9

Offset: 1

N. J. A. Sloane, Sep 17 2015

nonn

Cf. A261416, A260273.

A260273 Successively add the smallest nonzero binary number that is not a substring.

Original entry on oeis.org

1, 3, 5, 8, 11, 15, 17, 20, 23, 27, 31, 33, 36, 39, 44, 51, 56, 61, 65, 68, 71, 76, 81, 84, 87, 91, 95, 99, 104, 111, 115, 120, 125, 129, 132, 135, 140, 145, 148, 151, 157, 165, 168, 171, 175, 179, 186, 190, 194, 199, 204, 209, 216, 223, 227, 232, 241, 246

Offset: 1

Alex Meiburg, Jul 22 2015

a(n) is at least Omega(n), at most O(n*log(n)).
The empirical approximation n*(log(n)/2 + exp(1)) is startlingly close to tight, compared with many increasing upper bounds.
A261644(n) = A062383(a(n)) - a(n). - Reinhard Zumkeller, Aug 30 2015

			Begin with a(1)=1, in binary, "1". This contains the string "1" but not "10", so we add 2. Thus a(2)=1+2=3. This also contains "1" but not "10", so we move to a(3)=3+2=5. This contains "1" and "10" but not "11", so we add 3. Thus a(4)=5+3=8. (See A261018 for the successive numbers that are added. - _N. J. A. Sloane_, Aug 17 2015)

Alex Meiburg, Table of n, a(n) for n = 1..20000

See A261922 and A261461 for the smallest missing number function; also A261923, A262279, A261281.
Cf. A030308, A261015, A261016, A261017, A261019.
See also A261396 (when a(n) just passes a power of 2), A261416 (the limiting behavior just past a power of 2).
First differences are A261018.
A262288 is the decimal analog.
Cf. A261644, A062383.

a260273 n = a260273_list !! (n-1)
a260273_list = iterate (\x -> x + a261461 x) 1
-- Reinhard Zumkeller, Aug 30 2015, Aug 17 2015

public static void main(String[] args) {
   int a=1;
   for(int iter=0;iter<100;iter++){
       System.out.print(a+", ");
       int inc;
       for(inc=1; contains(a,inc); inc++);
       a+=inc;
   }
}
static boolean contains(int a,int test){
   int mask=(Integer.highestOneBit(test)<<1)-1;
   while(a >= test){
       if((a & mask) == test) return true;
       a >>= 1;
   }
   return false;
}

sublistQ[L1_, L2_] := Module[{l1 = Length[L1], l2 = Length[L2], k}, If[l2 <= l1, For[k = 1, k <= l1 - l2 + 1, k++, If[L1[[k ;; k + l2 - 1]] == L2, Return[True]]]]; False];
a[1] = 1; a[n_] := a[n] = Module[{bb = IntegerDigits[a[n-1], 2], k}, For[k = 1, sublistQ[bb, IntegerDigits[k, 2]], k++]; a[n-1] + k]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Apr 01 2016 *)
NestList[Function[k, k + FromDigits[#, 2] &@ SelectFirst[IntegerDigits[Range[2^8], 2], Length@ SequencePosition[IntegerDigits[k, 2], #] == 0 &]], 1, 64] (* Michael De Vlieger, Apr 01 2016, Version 10.1 *)

A260273_list, a = [1], 1
for i in range(10**3):
    b, s = 1, format(a,'b')
    while format(b,'b') in s:
        b += 1
    a += b
    s = format(a,'b')
    A260273_list.append(a) # Chai Wah Wu, Aug 26 2015

a(n+1) = a(n) + A261461(a(n)). - Reinhard Zumkeller, Aug 30 2015

A261644 Distance of A260273(n) to next power of 2.

Original entry on oeis.org

1, 1, 3, 8, 5, 1, 15, 12, 9, 5, 1, 31, 28, 25, 20, 13, 8, 3, 63, 60, 57, 52, 47, 44, 41, 37, 33, 29, 24, 17, 13, 8, 3, 127, 124, 121, 116, 111, 108, 105, 99, 91, 88, 85, 81, 77, 70, 66, 62, 57, 52, 47, 40, 33, 29, 24, 15, 10, 6, 2, 254, 251, 248, 245, 239

Offset: 1

Reinhard Zumkeller, Aug 30 2015

This sequence, as well as A261712, is suggested by A261396 and A261416.

			.  1: 1
.  2: 1
.  3: 3
.  4: 8,5,1
.  5: 15,12,9,5,1
.  6: 31,28,25,20,13,8,3
.  7: 63,60,57,52,47,44,41,37,33,29,24,17,13,8,3
.  8: 127,124,121,116,111,108,105,99,91,88,85,81,77,70,... (27 terms)
.  9: 254,251,248,245,239,236,233,227,218,213,207,202,195,,... (49 terms)

Reinhard Zumkeller, Rows n = 1..20 of triangle, flattened

Cf. A260273, A062383, A261645 (first differences), A261712 (reversed), A261646 (row lengths).

a261644 n = a261644_list !! (n-1)
a261644_list = zipWith (-)
               (map a062383 a260273_list) $ map fromIntegral a260273_list
a261644_tabf = [1] : f (tail $ zip a261645_list a261644_list) where
   f dxs = (map snd (dxs'' ++ [dx])) : f dxs' where
     (dxs'', dx:dxs') = span ((<= 0) . fst) dxs
a261644_row n = a261644_tabf !! (n-1)

a(n) = A062383(A260273(n)) - A260273(n).

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

A261727 Length of longest prefix of n-th row in A261712 coinciding with row n+1.

Original entry on oeis.org

1, 0, 0, 2, 0, 3, 0, 0, 0, 25, 46, 90, 172, 141, 275, 601, 1102, 2199, 4150

Offset: 1

Reinhard Zumkeller, Aug 31 2015

A261712(n+1,k) = A261712(n,k) for k = 1 .. a(n).

Cf. A261712, A261646, A261416.

a261727 n = a261727_list !! (n-1)
a261727_list = map (length . takeWhile (== 0)) $
                   zipWith (zipWith (-)) a261712_tabf $ tail a261712_tabf

A262290 a(0)=0; thereafter, add the smallest nonzero binary number, not a power of 2, that is not a substring of the binary expansion of a(n).

Original entry on oeis.org

0, 3, 8, 11, 17, 20, 23, 29, 38, 43, 49, 54, 61, 70, 75, 81, 84, 87, 93, 102, 107, 114, 119, 128, 131, 136, 139, 145, 148, 151, 157, 167, 173, 180, 187, 196, 201, 206, 211, 218, 225, 230, 235, 244, 253, 262, 267, 273, 276, 279, 285, 294, 299, 305, 310, 317, 327, 333, 340, 343, 349, 358, 365, 372, 381

Offset: 0

N. J. A. Sloane, Sep 19 2015

Cf. A260273, A262289.

Identical to A261416 except for first two terms.

a(n+1) = a(n) + A262289(a(n)).

cp's OEIS Frontend

A261926 First differences of A261416.

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A260273 Successively add the smallest nonzero binary number that is not a substring.

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A261644 Distance of A260273(n) to next power of 2.

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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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Mathematica

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A261727 Length of longest prefix of n-th row in A261712 coinciding with row n+1.

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Haskell

A262290 a(0)=0; thereafter, add the smallest nonzero binary number, not a power of 2, that is not a substring of the binary expansion of a(n).

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