A089648 -id:A089648 - cp's OEIS frontend

A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

1, -1, 0, -2, 1, -1, -1, -3, 2, 0, 0, -2, 0, -2, -2, -4, 3, 1, 1, -1, 1, -1, -1, -3, 1, -1, -1, -3, -1, -3, -3, -5, 4, 2, 2, 0, 2, 0, 0, -2, 2, 0, 0, -2, 0, -2, -2, -4, 2, 0, 0, -2, 0, -2, -2, -4, 0, -2, -2, -4, -2, -4, -4, -6, 5, 3, 3, 1, 3, 1, 1, -1, 3

Offset: 0

Clark Kimberling

-Sum_{n>=1} a(n)/((2*n)*(2*n+1)) = the "alternating Euler constant" log(4/Pi) = 0.24156... - (see A094640 and Sondow 2005, 2010).

a(A072600(n)) < 0; a(A072601(n)) <= 0; a(A031443(n)) = 0; a(A072602(n)) >= 0; a(A072603(n)) > 0; a(A031444(n)) = 1; a(A031448(n)) = -1; abs(a(A089648(n))) <= 1. - Reinhard Zumkeller, Feb 07 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).
Index entries for sequences related to binary expansion of n

Cf. A031443 for n when a(n)=0, A053738 for n when a(n) odd, A053754 for n when a(n) even, A030300 for a(n+1) mod 2.
Cf. A066879, A094640, A110625, A145037.
Cf. A072600, A072601, A072602, A072603, A031444, A031448, A089648.
See A268289 for a recurrence based on this sequence.

a037861 n = a023416 n - a000120 n  -- Reinhard Zumkeller, Aug 01 2013

A037861:= proc(n) local L;
     L:= convert(n,base,2);
     numboccur(0,L) - numboccur(1,L)
end proc:
map(A037861, [$0..100]); # Robert Israel, Mar 08 2016

Table[Count[ IntegerDigits[n, 2], 0] - Count[IntegerDigits[n, 2], 1], {n, 0, 75}]

a(n) = if (n==0, 1, 1 + logint(n, 2) - 2*hammingweight(n)); \\ Michel Marcus, May 15 2020 and Jun 16 2020

def A037861(n):
    return 2*format(n,'b').count('0')-len(format(n,'b')) # Chai Wah Wu, Mar 07 2016

From Henry Bottomley, Oct 27 2000: (Start)
a(n) = A023416(n) - A000120(n) = A029837(n) - 2*A000120(n) = 2*A023416(n) - A029837(n).
a(2*n) = a(n) + 1; a(2*n + 1) = a(2*n) - 2 = a(n) - 1. (End)
G.f. satisfies A(x) = (1 + x)*A(x^2) - x*(2 + x)/(1 + x). - Franklin T. Adams-Watters, Dec 26 2006
a(n) = b(n) for n > 0 with b(0) = 0 and b(n) = b(floor(n/2)) + (-1)^(n mod 2). - Reinhard Zumkeller, Dec 31 2007
G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(2^k)*(x^(2^k) - 1)/(1 + x^(2^k)). - Ilya Gutkovskiy, Apr 07 2018

A031448 Numbers whose base-2 representation has one fewer 0's than 1's.

Original entry on oeis.org

1, 5, 6, 19, 21, 22, 25, 26, 28, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 271, 279, 283, 285, 286, 295, 299, 301, 302, 307, 309, 310, 313, 314, 316, 327, 331, 333, 334, 339, 341, 342

Offset: 1

Clark Kimberling

A037861(a(n)) = -1. - Reinhard Zumkeller, Mar 31 2015

The viabin numbers of the integer partitions in which the number of parts is equal to the largest part (for the definition of viabin number see comment in A290253). For example, 99 is in the sequence because it is the viabin number of the integer partition [4,2,2,2]. - Emeric Deutsch, Aug 29 2017

			99 is in the sequence because its binary form is 1100011. - _Emeric Deutsch_, Aug 29 2017

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

Cf. A007088, A023416, A000120, A031444, subsequence of A089648.

a031448 n = a031448_list !! (n-1)
a031448_list = filter ((== -1) . a037861) [1..]
-- Reinhard Zumkeller, Mar 31 2015

vitopart := proc (n) local L, i, j, N, p, t: N := 2*n; L := ListTools:-Reverse(convert(N, base, 2)): j := 0: for i to nops(L) do if L[i] = 0 then j := j+1: p[j] := numboccur(L[1 .. i], 1) end if end do: sort([seq(p[t], t = 1 .. j)], `>=`) end proc: A := {}; for m to 500 do if nops(vitopart(m)) = max(vitopart(m)) then A := `union`(A, {m}) else  end if end do: A; # program is based on my comment; the command vitopart(n) yields the integer partition having viabin number n. - Emeric Deutsch, Aug 29 2017

Select[Range[400],DigitCount[#,2,1]==DigitCount[#,2,0]+1&] (* Harvey P. Dale, May 24 2019 *)

A031444 Numbers whose base-2 representation has one more 0 than 1's.

Original entry on oeis.org

4, 17, 18, 20, 24, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 263, 267, 269, 270, 275, 277, 278, 281, 282, 284, 291, 293, 294, 297, 298, 300, 305, 306, 308, 312, 323, 325, 326, 329, 330, 332, 337, 338, 340, 344

Offset: 1

Clark Kimberling

If m is a term, then also 4*m+1. - Reinhard Zumkeller, Mar 31 2015

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

Cf. A007088, A023416, A000120, A031448, A037861, A095072 (subsequence).

Subsequence of A089648.

a031444 n = a031444_list !! (n-1)
a031444_list = filter ((== 1) . a037861) [1..]
-- Reinhard Zumkeller, Mar 31 2015

Select[Range[350], (Differences@ DigitCount[#, 2])[[1]] == 1 &] (* Amiram Eldar, Aug 03 2023 *)

A037861(a(n)) = 1. - Reinhard Zumkeller, Mar 31 2015

A343258 Numbers whose binary representation has a prime number of zeros and a prime number of ones.

Original entry on oeis.org

9, 10, 12, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 65, 66, 68, 72, 79, 80, 87, 91, 93, 94, 96, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 131, 133, 134, 137, 138, 140, 143, 145, 146, 148, 151, 152, 155, 157, 158

Offset: 1

Jean-Jacques Vaudroz, Apr 09 2021

Terms of 4, 5 and 6 total bits (9 through 56) are the same as A089648.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 78 terms from Jean-Jacques Vaudroz)

Intersection of A052294 and A144754.

Cf. A089648.

q:= n->(l->(t->andmap(isprime, [t, nops(l)-t]))(add(i, i=l)))(Bits[Split](n)):
select(q, [$1..200])[];  # Alois P. Heinz, Apr 11 2021

Select[Range[160], And @@ PrimeQ[DigitCount[#, 2]] &] (* Amiram Eldar, Apr 09 2021 *)

isa(n)= isprime(hammingweight(n));
isb(n)= isprime(#binary(n) - hammingweight(n));
isok(n) = isa(n) && isb(n);

from sympy import isprime
def ok(n): b = bin(n)[2:]; return all(isprime(b.count(d)) for d in "01")
print(list(filter(ok, range(159)))) # Michael S. Branicky, Sep 10 2021

cp's OEIS Frontend

A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

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A031448 Numbers whose base-2 representation has one fewer 0's than 1's.

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A031444 Numbers whose base-2 representation has one more 0 than 1's.

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A343258 Numbers whose binary representation has a prime number of zeros and a prime number of ones.

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Maple

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Python