A087118 -id:A087118 - cp's OEIS frontend

A087116 Number of maximal groups of consecutive zeros in binary representation of n.

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

Offset: 0

Reinhard Zumkeller, Aug 14 2003

nonn

The following four statements are equivalent: a(n) = 0; n = 2^k - 1 for some k; A087117(n) = 0; A023416(n) = 0.

			G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^8 + x^9 + 2*x^10 + x^11 + x^12 + x^13 + x^14 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Cf. A087118, A087119, A087120, A023416, A007088.
Cf. A023416, A087117.
Cf. A033264, A003817, A069010.
Essentially the same as A033264.

a087116 0 = 1
a087116 n = f 0 n where
   f y 0 = y
   f y x = if r == 0 then g x' else f y x'
           where (x', r) = divMod x 2
                 g z = if r == 0 then g z' else f (y + 1) z'
                       where (z', r) = divMod z 2
-- Reinhard Zumkeller, Mar 31 2015

a[n_] := SequenceCount[IntegerDigits[n, 2], {Longest[0..]}];
Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Oct 18 2021 *)

a(n) = if (n == 0, 1, hammingweight(bitxor(n, n>>1)) >> 1);
vector(102, i, a(i-1))  \\ Gheorghe Coserea, Sep 17 2015

def A087116(n):
    return sum(1 for d in bin(n)[2:].split('1') if len(d)) # Chai Wah Wu, Nov 04 2016

a(n) = A033264(n) for n > 0 since strings of 0's alternate with strings of 1's. - Jonathan Sondow, Jan 17 2016
a(n) = a(2*n + 1) = a(4*n + 2) - 1, if n > 0. - Michael Somos, Nov 04 2016
a(n) = A069010(A003817(n)-n) for n > 0. - Chai Wah Wu, Nov 04 2016

A087120 Smallest numbers having in binary representation exactly n maximal groups of consecutive zeros.

Original entry on oeis.org

1, 0, 10, 42, 170, 682, 2730, 10922, 43690, 174762, 699050, 2796202, 11184810, 44739242, 178956970, 715827882, 2863311530, 11453246122, 45812984490, 183251937962, 733007751850, 2932031007402, 11728124029610, 46912496118442

Offset: 0

Reinhard Zumkeller, Aug 14 2003

nonn

A087116(a(n))=n and A087116(k)

For n>1, a(n) = A020988(n-1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for sequences related to binary expansion of n

Cf. A087118, A023416, A007088.

Join[{1,0},NestList[4#+2&,10,25]] (* Harvey P. Dale, Apr 20 2019 *)

a(0)=1, a(1)=0, a(2)=10, a(n)=4*a(n-1)+2.

A087119 Numbers having more than one maximal group of consecutive zeros in binary representation of n.

Original entry on oeis.org

10, 18, 20, 21, 22, 26, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 50, 52, 53, 54, 58, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 98, 100, 101, 102, 104, 105, 106, 107, 108, 109, 110, 114, 116, 117, 118, 122

Offset: 1

Reinhard Zumkeller, Aug 14 2003

nonn

A087116(a(n)) > 1.

Index entries for sequences related to binary expansion of n

Cf. A087118, A023416, A007088.

A356647 Concatenation of runs {y..x} for each x>=1, using each y from 1 to x before moving on to the next value for x.

Original entry on oeis.org

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

Offset: 1

Jonathan Kal-El Peréz, Aug 19 2022

Alternate definition: Flattened list of all suffixes (ordered longest to shortest) of the list of all prefixes (ordered shortest to longest) of the list of positive integers. A prefix here is defined as any contiguous sublist of a list which includes the first element, and a suffix as any contiguous sublist of a list which includes the last element.

Also, concatenation of runs A002260(n)..A002024(n) for each n>=1.

The Nineteenth Byte Stack Exchange chat room, Message regarding this sequence. Some replies are programs to generate terms of the sequence.

Cf. A000120, A004006, A087118.

a=n=>{for(let i=1;++i;){for(let j=0;++j

function a = A356647( max_x )
    a = cell2mat(arrayfun(@(x)(cell2mat(arrayfun(@(y)([y:x]),[1:x],'UniformOutput', false))) ...
        ,[1:max_x],'UniformOutput', false));
end % Thomas Scheuerle, Sep 30 2022

Print @ Flatten @ (Reverse@FoldList[Join[#2,#]&, {#}&/@Reverse@#]& /@ FoldList[Join, Table[{n},{n,1,10}]])

from itertools import count, islice
def agen(): # generator of terms
    for k in count(1):
        for j in range(1, k+1):
            yield from range(j, k+1)
print(list(islice(agen(), 87))) # Michael S. Branicky, Oct 11 2022

a(n) = A000120(A087118(n + 1)). - Thomas Scheuerle, Feb 16 2023

a(n*(n^2 + 5)/6) = a(A004006(n)) = n. This is the earliest appearance of n. - Thomas Scheuerle, Sep 30 2022

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A087116 Number of maximal groups of consecutive zeros in binary representation of n.

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A087120 Smallest numbers having in binary representation exactly n maximal groups of consecutive zeros.

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A087119 Numbers having more than one maximal group of consecutive zeros in binary representation of n.

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A356647 Concatenation of runs {y..x} for each x>=1, using each y from 1 to x before moving on to the next value for x.

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