A260273 -id:A260273 - cp's OEIS frontend

A261416 Let b(k) denote A260273(k). It appears that for k >= 200, whenever b(k) just passes a power of 2, 2^m say, the successive differences b(k)-2^m converge to this sequence.

2, 5, 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, Aug 25 2015

nonn

It would be nice to have an independent characterization of this sequence.

A partial answer: set a(0)=2, and for n>0, a(n) = A261281(a(n-1)). - N. J. A. Sloane, Sep 17 2015

			At k=200, b(k)=b(200)=1026 has just passed 2^10. The successive differences b(200+i)-2^10 (i>=0) beyond this point are 2, 5, 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, [165, ...], which are the first 31 terms of the present sequence.
At k=371, b(371)=2050, and the successive differences b(371+i)-2^11 are 2, 5, ..., 279, 285, ... giving the first 51 terms of the present sequence.

Cf. A260273, A261281. For when A260273 just passes a power of 2, see A261396.

A261018 First differences of A260273.

Original entry on oeis.org

2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 2, 3, 3, 5, 7, 5, 5, 4, 3, 3, 5, 5, 3, 3, 4, 4, 4, 5, 7, 4, 5, 5, 4, 3, 3, 5, 5, 3, 3, 6, 8, 3, 3, 4, 4, 7, 4, 4, 5, 5, 5, 7, 7, 4, 5, 9, 5, 4, 4, 4, 3, 3, 3, 6, 3, 3, 6, 9, 5, 6, 5, 7, 8, 3, 3, 6, 3, 3, 4, 4, 4, 7, 4, 4, 8, 4, 4, 5, 5

Offset: 1

N. J. A. Sloane, Aug 17 2015

nonn

a(n) = A261461(A260273(n)). - Reinhard Zumkeller, Aug 30 2015

N. J. A. Sloane, Table of n, a(n) for n = 1..19999

Cf. A260273.

Cf. A261645, A261461.

a261018 n = a261018_list !! (n-1)
a261018_list = zipWith (-) (tail a260273_list) a260273_list
-- Reinhard Zumkeller, Aug 30 2015

b[1] = 1;
b[n_] := b[n] = Module[{bits, k}, bits = IntegerDigits[b[n-1], 2]; For[k = 1, True, k++, If[SequencePosition[bits, IntegerDigits[k, 2]] == {}, Return[b[n-1] + k]]]];
a[n_] := b[n+1] - b[n];
Array[a, 100] (* Jean-François Alcover, Aug 02 2018 *)

A261018_list, a = [], 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')
    A261018_list.append(b) # Chai Wah Wu, Aug 26 2015

A261396 a(n) = smallest k such that A260273(k) >= 2^n.

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 19, 34, 61, 110, 200, 371, 697, 1310, 2484, 4739, 9072, 17458, 33671, 65128, 126225, 244802, 475124, 922891, 1793461, 3487348, 6784691, 13208038, 25731600, 50166771, 97873783, 191089176, 373349780, 729972649, 1428257200, 2796453078, 5478981032, 10741710906, 21072415837

Offset: 0

N. J. A. Sloane, Aug 17 2015

nonn

The sequence indicates the first time a term in A260273 has binary length n+1.

A261646 = first differences = row lengths of tables A261644 and A261712. - Reinhard Zumkeller, Aug 30 2015

Cf. A260273.

Cf. A261646, A261644, A261712.

a261396 n = a261396_list !! (n-1)
a261396_list = f 1 1 a260273_list where
   f z k (x:xs) | x >= z    = k : f (2 * z) (k + 1) xs
                | otherwise = f z (k + 1) xs
-- Reinhard Zumkeller, Aug 30 2015

a(18)-a(23) from Alois P. Heinz, Aug 19 2015
a(24)-a(34) from Chai Wah Wu, Aug 26 2015
a(35)-a(38) from Chai Wah Wu, Aug 31 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).

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

Original entry on oeis.org

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

Reinhard Zumkeller, Aug 30 2015

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

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

Cf. A007088, A030308, A260273, A261018; record values and where they occur: A261466, A261467.
See A261922 for a variant.
Cf. A062289, A144016, A261922, A126646.

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

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 *)

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

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

A261015 Irregular triangle read by rows: T(n,k) (0 <= k <= 2^n-1) = number of binary strings of length n such that the smallest number whose binary representation is not visible in the string is k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 0, 0, 0, 1, 1, 3, 6, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 4, 11, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 19, 21, 15, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Aug 16 2015

Suggested by A260273.

			Triangle begins:
  1,1,
  1,1,1,1,
  1,1,2,3,1,0,0,0,
  1,1,3,6,4,1,0,0,0,0,0,0,0,0,0,0,
...
For row 3, here are the 8 strings of length 3 and for each one, the smallest missing number k:
 000 1
 001 2
 010 3
 011 2
 100 3
 101 3
 110 4
 111 0

Alois P. Heinz, Rows n = 1..15, flattened

Cf. A260273, A261016, A261017.

See A261019 for a more compact version (which has further information about the columns).

notVis[bits_] := For[i = 0, True, i++, If[SequencePosition[bits, IntegerDigits[i, 2]] == {}, Return[i]]];
T[n_, k_] := Select[Rest[IntegerDigits[#, 2]]& /@ Range[2^n, 2^(n+1)-1], notVis[#] == k&] // Length;
Table[T[n, k], {n, 1, 6}, {k, 0, 2^n-1}] // Flatten (* Jean-François Alcover, Aug 02 2018 *)

More terms from Alois P. Heinz, Aug 17 2015

A261016 a(n) = Sum_{k=0..2^n-1} k*A261015(n,k).

Original entry on oeis.org

1, 6, 18, 46, 107, 241, 535, 1178, 2569, 5546, 11859, 25156, 53058, 111379, 232966, 486023, 1012185, 2104729, 4370644, 9064924, 18778766, 38856079, 80307630, 165790125, 341872016, 704171185, 1448812630, 2977673003, 6113469501, 12538958895, 25693167881, 52598980642

Offset: 1

N. J. A. Sloane, Aug 16 2015

nonn

The scaled values a(n)/2^n are (to nine decimal places) 0.5000000000, 1.500000000, 2.250000000, 2.875000000, 3.343750000, 3.765625000, 4.179687500, 4.601562500, 5.017578125, 5.416015625, 5.790527344, 6.141601562, 6.476806641, 6.798034668, 7.109558105, 7.416122437, 7.722358704, 8.028903961, 8.336341858, 8.644985199, 8.954413414, 9.264011145, 9.573415518, 9.881861508, 10.18858004, 10.49296834, 10.79449527, 11.09269635, 11.38722431, 11.67781548, 11.96431363, 12.24665452, ...

Alois P. Heinz and Hiroaki Yamanouchi, Table of n, a(n) for n = 1..58 (a(1)-a(36) from Alois P. Heinz)

Cf. A261019, A261015, A260273.

a261016 = sum . zipWith (*) [0..] . a261019_row'
-- Reinhard Zumkeller, Aug 18 2015

(* This program is not suitable to compute more than a dozen terms. *)
notVis[bits_] := For[i = 0, True, i++, If[SequencePosition[bits, IntegerDigits[i, 2]] == {}, Return[i]]];
T[n_, k_] := Select[Rest[IntegerDigits[#, 2]] & /@ Range[2^n, 2^(n+1) - 1], notVis[#] == k &] // Length;
a[n_] := Sum[k*T[n, k], {k, 0, 2^n - 1}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Aug 02 2018 *)

a(5)-a(16) from Alois P. Heinz, Aug 17 2015
a(17)-a(25) from Reinhard Zumkeller, Aug 18 2015
a(26)-a(32) from Alois P. Heinz, Aug 19 2015

A261017 a(n) = max k such that A261015(n,k) is not zero.

Original entry on oeis.org

1, 3, 4, 5, 5, 7, 8, 9, 9, 9, 11, 11, 13, 15, 16, 17, 17, 17, 17, 19, 19, 19, 21, 21, 23, 23, 23, 27, 29, 31, 32, 33, 33, 33, 33, 33, 35, 35, 35, 35, 37, 37, 37, 39, 39, 39, 39, 41, 41, 43, 43, 45, 45, 45, 47, 47, 47, 47

Offset: 1

N. J. A. Sloane, Aug 17 2015

Cf. A261019, A261015, A261016, A260273.

a261017 = subtract 1 . length . a261019_row
-- Reinhard Zumkeller, Aug 18 2015

(* This program is not suitable to compute more than a dozen terms. *)
notVis[bits_] := For[i = 0, True, i++, If[SequencePosition[bits, IntegerDigits[i, 2]] == {}, Return[i]]];
T[n_, k_] := Select[Rest[IntegerDigits[#, 2]] & /@ Range[2^n, 2^(n+1) - 1], notVis[#] == k &] // Length;
a[n_] := Do[If[T[n, k] > 0, Return[k]], {k, 2^n - 1, 0, -1}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Aug 02 2018 *)

a(5)-a(17) from Alois P. Heinz, Aug 17 2015
a(18)-a(25) from Reinhard Zumkeller, Aug 18 2015
a(26)-a(36) from Alois P. Heinz, Aug 19 2015
a(37)-a(58) from Hiroaki Yamanouchi, Aug 23 2015

A261786 Successively add the smallest ternary number that is not a substring.

Original entry on oeis.org

1, 3, 5, 8, 9, 11, 15, 18, 19, 22, 25, 28, 30, 32, 36, 38, 43, 46, 49, 52, 55, 58, 61, 64, 68, 71, 74, 75, 79, 82, 84, 86, 90, 92, 96, 100, 104, 108, 110, 115, 120, 122, 125, 128, 131, 134, 137, 140, 143, 146, 150, 153, 156, 160, 163, 166, 169, 172, 176, 179

Offset: 1

Reinhard Zumkeller, Sep 01 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007089, A261787, A261789 (first differences), A261788, A260273, A261793.

a261786 n = a261786_list !! (n-1)
a261786_list = iterate (\x -> x + a261787 x) 1

ts(n) = Str(fromdigits(digits(n, 3))); \\ A007089
f(n) = my(s=ts(n), k=1); while (#strsplit(s, ts(k)) != 1, k++); k; \\ A261787
lista(nn) = my(v = vector(nn)); v[1] = 1; for (n=2, nn, v[n] = v[n-1] + f(v[n-1]);); v; \\ Michel Marcus, Feb 06 2022

A261793 Successively add the smallest number that is not a substring in decimal representation.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 17, 19, 21, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Reinhard Zumkeller, Sep 01 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A031298, A261794, A261795 (first differences), A261806, A260273, A261786

a261793 n = a261793_list !! (n-1)
a261793_list = iterate (\x -> x + a261794 x) 1

cp's OEIS Frontend

A261416 Let b(k) denote A260273(k). It appears that for k >= 200, whenever b(k) just passes a power of 2, 2^m say, the successive differences b(k)-2^m converge to this sequence.

Views

Author

Keywords

Comments

Examples

Crossrefs

A261018 First differences of A260273.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Python

A261396 a(n) = smallest k such that A260273(k) >= 2^n.

Views

Author

Keywords

Comments

Crossrefs

Programs

Haskell

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Formula

A261015 Irregular triangle read by rows: T(n,k) (0 <= k <= 2^n-1) = number of binary strings of length n such that the smallest number whose binary representation is not visible in the string is k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A261016 a(n) = Sum_{k=0..2^n-1} k*A261015(n,k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A261017 a(n) = max k such that A261015(n,k) is not zero.

Views

Author

Keywords

Crossrefs

Programs

Haskell