A261461 -id:A261461 - cp's OEIS frontend

A261467 Where records occur in A261461.

0, 1, 2, 6, 12, 44, 92, 184, 1208, 1256, 4792, 9912, 19832, 39664, 563952, 576464, 4496112, 4499184, 17996528, 17997488, 143972080, 145057520, 145070832, 294967024, 589944560, 1179889136, 2359778272, 71079255008, 72019201952

Offset: 1

Reinhard Zumkeller, Aug 30 2015

A261461(a(n)) = A261466(n).

Rémy Sigrist, C program

Cf. A261461, A261466.

C
```
See Links section.
```

a261467 n = a261467_list !! (n-1)
-- where a261467_list is defined in A261466.

from itertools import count, islice
def f(n):
    b, k = bin(n)[2:], 1
    return next(k for k in count(1) if bin(k)[2:] not in b)
def agen(record=-1):
    yield from (((record:=f(k)), k)[1] for k in count(0) if f(k) > record)
print(list(islice(agen(), 18))) # Michael S. Branicky, Feb 26 2023

Leading zero and more terms from Rémy Sigrist, Mar 10 2018

a(26)-a(29) from Rémy Sigrist, Feb 26 2023

A261466 Records in A261461.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35

Offset: 1

Reinhard Zumkeller, Aug 30 2015

a(n) = A261461(A261467(n)).

Cf. A261461, A261467.

a261466 n = a261466_list !! (n-1)
(a261466_list, a261467_list) = unzip $ (0, 1) : f 0 1 where
   f i x | y > x     = (y, i) : f (i + 1) y
         | otherwise = f (i + 1) x
         where y = a261461 i

from itertools import count, islice
def f(n):
    b, k = bin(n)[2:], 1
    return next(k for k in count(1) if bin(k)[2:] not in b)
def agen(record=-1):
    yield from (((record:=f(k)), k)[0] for k in count(0) if f(k) > record)
print(list(islice(agen(), 18))) # Michael S. Branicky, Feb 26 2023

Initial terms corrected and more terms from Rémy Sigrist, Mar 10 2018

a(26)-a(29) from Rémy Sigrist, Feb 26 2023

A126646 a(n) = 2^(n+1) - 1.

Original entry on oeis.org

1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647

Offset: 0

Aleksandar M. Janjic and Milan Janjic, Feb 08 2007, Feb 13 2007

a(n) is the number of integers k less than 10^n such that the decimal representation of k lacks the digits 1,2,3,4,5,6 and 7 and at least one of the digits 8,9.
Partial sums of the powers of 2 (A000079).
a(n) is the number of elements (all m-dimensional faces) in an n-dimensional simplex (0 <= m <= n). - Sergey Pavlov, Aug 15 2015
A261461(a(n)) != A261922(a(n)). - Reinhard Zumkeller, Sep 17 2015
a(n) is the total number of matches in a knockout tournament with 2^n players. - Paul Duckett, Dec 12 2022

			a(8) = 2^9 - 1 = 511.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Wikipedia, Simplex Elements (see last column of table).
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Essentially the same as A000225.
Cf. A125630, A125945, A125947, A125948, A125940, A125909, A125908, A125880, A125897, A125904, A125858.
Cf. A000079, A168604.
Cf. A261461, A261922.

a126646 = (subtract 1) . (2 ^) . (+ 1)
a126646_list = iterate ((+ 1) . (* 2)) 1
-- Reinhard Zumkeller, Sep 17 2015

[2^(n+1)-1: n in [0.. 35]]; // Vincenzo Librandi, Aug 20 2015

A126646:=n->2*2^n-1; seq(A126646(n), n=0..50); # Wesley Ivan Hurt, Dec 02 2013

Table[2^(n+1) - 1, {n, 0, 50}] (* Wesley Ivan Hurt, Dec 02 2013 *)
LinearRecurrence[{3,-2},{1,3},40] (* Harvey P. Dale, Mar 23 2018 *)

first(m)=vector(m,i,i--;2^(i+1)-1) /* Anders Hellström, Aug 19 2015 */

def A126646(n): return (1<Chai Wah Wu, Mar 18 2024

[2^(n+1) -1 for n in (0..50)] # G. C. Greubel, Mar 31 2021

a(n-1)^2 + a(n) = a(2n) + 1, a square. - Vincenzo Librandi and Ralf Stephan, Nov 23 2010
G.f.: 1/ ( (1-2*x)*(1-x) ). - R. J. Mathar, Dec 02 2013
a(n) = 3*a(n-1) - 2*a(n-2), n > 1. - Wesley Ivan Hurt, Aug 21 2015
E.g.f.: 2*exp(2*x) - exp(x). - G. C. Greubel, Mar 31 2021

A062289 Numbers n such that n-th row in Pascal triangle contains an even number, i.e., A048967(n) > 0.

Original entry on oeis.org

2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 02 2001

nonn

Numbers n such that binary representation contains the bit string "10". Union of A043569 and A101082. - Rick L. Shepherd, Nov 29 2004

The asymptotic density of this sequence is 1 (Burns, 2016). - Amiram Eldar, Jan 26 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10001
Rob Burns, Asymptotic density of Catalan numbers modulo 3 and powers of 2, arXiv:1611.03705 [math.NT], 2016.
Oliver Kullmann and Xishun Zhao, On variables with few occurrences in conjunctive normal forms, in: K. A. Sakallah and L. Simon (eds), International Conference on Theory and Applications of Satisfiability Testing, Springer, Berlin, Heidelberg, 2011, pp. 33-46; arXiv preprint, arXiv:1010.5756 [cs.DM], 2010-2011.
Oliver Kullmann and Xishun Zhao, Parameters for minimal unsatisfiability: Smarandache primitive numbers and full clauses, arXiv preprint, arXiv:1505.02318 [cs.DM], 2015.

Complement of A000225, so these might be called non-Mersenne numbers.
A132782 is a subsequence.
Cf. A048967, A057716, A043569, A101082.
Cf. A007461, A036987, A102370, A105027.
Cf. A261461, A261922.

a062289 n = a062289_list !! (n-1)
a062289_list = 2 : g 2 where
   g n = nM n : g (n+1)
   nM k = maximum $ map (\i -> i + min i (a062289 $ k-i+1)) [2..k]
   -- Cf. link [Oliver Kullmann, Xishun Zhao], Def. 3.1, page 3.
-- Reinhard Zumkeller, Feb 21 2012, Dec 31 2010

ok[n_] := MatchQ[ IntegerDigits[n, 2], {_, 1, 0, _}]; Select[ Range[100], ok] (* Jean-François Alcover, Dec 12 2011, after Rick L. Shepherd *)

isok(m) = #select(x->((x%2)==0), vector(m+1, k, binomial(m, k-1))); \\ Michel Marcus, Jan 26 2021

def A062289(n): return n+(m:=n.bit_length())-(not n>=(1<Chai Wah Wu, Jun 30 2024

a(n) = A057716(n+1) - 1.
a(n) = 2 if n=1, otherwise max{min{2*i, a(n-i+1) + i}: 1 < i <= n}.
A036987(a(n)) = 0. - Reinhard Zumkeller, Mar 06 2012
A007461(a(n)) mod 2 = 0. - Reinhard Zumkeller, Apr 02 2012
A102370(n) = A105027(a(n)). - Reinhard Zumkeller, Jul 21 2012
A261461(a(n)) = A261922(a(n)). - Reinhard Zumkeller, Sep 17 2015

More terms from Rick L. Shepherd, Nov 29 2004

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

A261922 a(n) = smallest nonnegative number that is not a substring of n in its binary representation.

Original entry on oeis.org

1, 0, 3, 0, 3, 3, 4, 0, 3, 3, 3, 4, 5, 4, 4, 0, 3, 3, 3, 5, 3, 3, 4, 4, 5, 5, 4, 4, 5, 4, 4, 0, 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, 0, 3, 3, 3, 5, 3, 3, 5, 5, 3, 3, 3, 6, 5, 7, 5, 5, 3, 3, 3, 6, 3, 3, 4, 4, 7, 7, 4, 4, 8, 4, 4, 4, 5, 5, 5, 5, 5, 7

Offset: 0

N. J. A. Sloane, Sep 16 2015

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

Similar to A261461.

Cf. A007088, A030308, A062289, A126646, A262281.

import Data.List (isInfixOf)
a261922 x = f a030308_tabf where
   f (cs:css) = if isInfixOf cs (a030308_row x)
                   then f css else foldr (\d v -> 2 * v + d) 0 cs
-- Reinhard Zumkeller, Sep 17 2015

bstr(n) = if (n==0, "0", my(s="", b=binary(n)); for (i=1, #b, s=concat(s, b[i])); s);
a(n) = my(sn=btostr(n), k=0); while (#strsplit(sn, bstr(k)) != 1, k++); k; \\ Michel Marcus, Sep 20 2023

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

From Reinhard Zumkeller, Sep 17 2015: (Start)
a(A062289(n)) = A261461(A062289(n)).
a(A126646(n)) != A261461(A126646(n)). (End)

A144016 a(n) = the largest positive integer m such that the binary representations of all positive integers <= m are found within the binary representation of n.

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Sep 07 2008

From Rémy Sigrist, Mar 10 2018: (Start)
a(n) is the greatest k <= n such that A213629(n, i) > 0 for i = 1..k.
See A261467 for the indices of record values.
(End)

			44 in binary is 101100. In this string we find 1 (1 in decimal): (1)01100; 10 (2 in decimal): (10)1100; 11 (3 in decimal): 10(11)00; 100 (4 in decimal): 101(100); 101 (5 in decimal): (101)100; and 110 (6 in decimal): 10(110)0; but not 111 (7 in decimal). So a(44) = 6.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A213629, A261461, A261467.

a(n) = A261461(n) - 1. - Rémy Sigrist, Mar 10 2018

Extended by Ray Chandler, Nov 07 2008

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

A261787 a(n) is the smallest nonzero number that is not a substring of n in ternary representation.

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 4, 2, 2, 3, 3, 3, 3, 1, 3, 1, 4, 3, 3, 1, 3, 1, 2, 2, 4, 2, 2, 4, 4, 4, 4, 2, 2, 5, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 1, 3, 1, 4, 3, 3, 1, 3, 1, 4, 4, 4, 5, 3, 3, 3, 3, 3, 1, 3, 1, 4, 3, 3, 1, 3, 1, 2, 2, 4, 2, 2

Offset: 0

Reinhard Zumkeller, Sep 01 2015

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

Cf. A007089, A030341, A261789, A261786, A261461, A261794.

import Data.List (isInfixOf)
a261787 x = f $ tail a030341_tabf where
   f (cs:css) = if isInfixOf cs (a030341_row x)
                   then f css else foldr (\d v -> 3 * v + d) 0 cs

ts(n) = Str(fromdigits(digits(n, 3)));
a(n) = my(s=ts(n), k=1); while (#strsplit(s, ts(k)) != 1, k++); k; \\ Michel Marcus, Feb 05 2022

A261789(n) = a(A261786(n)).

A261789 First differences of A261786.

Original entry on oeis.org

2, 2, 3, 1, 2, 4, 3, 1, 3, 3, 3, 2, 2, 4, 2, 5, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 1, 4, 3, 2, 2, 4, 2, 4, 4, 4, 4, 2, 5, 5, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3, 1, 4, 3, 4, 4, 5, 3, 3, 3, 3, 3, 1, 4, 3, 4, 3, 3, 1, 4, 3, 2, 2, 4, 2, 4, 4

Offset: 1

Reinhard Zumkeller, Sep 01 2015

nonn

a(n) = A261787(A261786(n)).

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

Cf. A261645, A261461, A261786, A261018, A261795.

a261789 n = a261789_list !! (n-1)
a261789_list = zipWith (-) (tail a261786_list') a261786_list'

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A261467 Where records occur in A261461.

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A261466 Records in A261461.

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A126646 a(n) = 2^(n+1) - 1.

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A062289 Numbers n such that n-th row in Pascal triangle contains an even number, i.e., A048967(n) > 0.

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

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A261922 a(n) = smallest nonnegative number that is not a substring of n in its binary representation.

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A144016 a(n) = the largest positive integer m such that the binary representations of all positive integers <= m are found within the binary representation of n.