A031448 -id:A031448 - 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

A345910 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.

Original entry on oeis.org

6, 20, 25, 27, 30, 72, 81, 83, 86, 92, 98, 101, 103, 106, 109, 111, 116, 121, 123, 126, 272, 289, 291, 294, 300, 312, 322, 325, 327, 330, 333, 335, 340, 345, 347, 350, 360, 369, 371, 374, 380, 388, 393, 395, 398, 402, 405, 407, 410, 413, 415, 420, 425, 427

Offset: 1

Gus Wiseman, Jul 01 2021

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of terms together with the corresponding compositions begins:
      6: (1,2)
     20: (2,3)
     25: (1,3,1)
     27: (1,2,1,1)
     30: (1,1,1,2)
     72: (3,4)
     81: (2,4,1)
     83: (2,3,1,1)
     86: (2,2,1,2)
     92: (2,1,1,3)
     98: (1,4,2)
    101: (1,3,2,1)
    103: (1,3,1,1,1)
    106: (1,2,2,2)
    109: (1,2,1,2,1)

These compositions are counted by A001791.
A version using runs of binary digits is A031444.
These are the positions of -1's in A124754.
The opposite (positive 1) version is A345909.
The reverse version is A345912.
The version for alternating sum of prime indices is A345959.
Standard compositions: A000120, A066099, A070939, A124754, A228351, A344618.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions of 2n+1 with alternating sum 1, ranked by A001105.
A011782 counts compositions.
A097805 counts compositions by sum and alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000097, A000346, A008549, A025047, A027187, A031443, A031448, A114121, A119899, A126869, A238279, A344617.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]==-1&]

A345911 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum 1.

Original entry on oeis.org

1, 6, 7, 20, 21, 26, 27, 30, 31, 72, 73, 82, 83, 86, 87, 92, 93, 100, 101, 106, 107, 110, 111, 116, 117, 122, 123, 126, 127, 272, 273, 290, 291, 294, 295, 300, 301, 312, 313, 324, 325, 330, 331, 334, 335, 340, 341, 346, 347, 350, 351, 360, 361, 370, 371, 374

Offset: 1

Gus Wiseman, Jul 01 2021

nonn

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of terms together with the corresponding compositions begins:
     1: (1)
     6: (1,2)
     7: (1,1,1)
    20: (2,3)
    21: (2,2,1)
    26: (1,2,2)
    27: (1,2,1,1)
    30: (1,1,1,2)
    31: (1,1,1,1,1)
    72: (3,4)
    73: (3,3,1)
    82: (2,3,2)
    83: (2,3,1,1)
    86: (2,2,1,2)
    87: (2,2,1,1,1)

These compositions are counted by A000984 (bisection of A126869).
The version for Heinz numbers of partitions is A001105.
A version using runs of binary digits is A066879.
These are positions of 1's in A344618.
The non-reverse version is A345909.
The opposite (negative 1) version is A345912.
The version for prime indices is A345958.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating or reverse-alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A008549, A025047, A027193, A031448, A034871, A114121, A120452, A344607.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]==1&]

A345909 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 1.

Original entry on oeis.org

1, 5, 7, 18, 21, 23, 26, 29, 31, 68, 73, 75, 78, 82, 85, 87, 90, 93, 95, 100, 105, 107, 110, 114, 117, 119, 122, 125, 127, 264, 273, 275, 278, 284, 290, 293, 295, 298, 301, 303, 308, 313, 315, 318, 324, 329, 331, 334, 338, 341, 343, 346, 349, 351, 356, 361

Offset: 1

Gus Wiseman, Jun 30 2021

nonn

The alternating sum of a composition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence of terms together with the corresponding compositions begins:
      1: (1)             87: (2,2,1,1,1)
      5: (2,1)           90: (2,1,2,2)
      7: (1,1,1)         93: (2,1,1,2,1)
     18: (3,2)           95: (2,1,1,1,1,1)
     21: (2,2,1)        100: (1,3,3)
     23: (2,1,1,1)      105: (1,2,3,1)
     26: (1,2,2)        107: (1,2,2,1,1)
     29: (1,1,2,1)      110: (1,2,1,1,2)
     31: (1,1,1,1,1)    114: (1,1,3,2)
     68: (4,3)          117: (1,1,2,2,1)
     73: (3,3,1)        119: (1,1,2,1,1,1)
     75: (3,2,1,1)      122: (1,1,1,2,2)
     78: (3,1,1,2)      125: (1,1,1,1,2,1)
     82: (2,3,2)        127: (1,1,1,1,1,1,1)
     85: (2,2,2,1)      264: (5,4)

These compositions are counted by A000984 (bisection of A126869).
The version for prime indices is A001105.
A version using runs of binary digits is A031448.
These are the positions of 1's in A124754.
The opposite (negative 1) version is A345910.
The reverse version is A345911.
The version for Heinz numbers of partitions is A345958.
Standard compositions: A000120, A066099, A070939, A124754, A228351, A344618.
A000070 counts partitions with alternating sum 1 (ranked by A345957).
A000097 counts partitions with alternating sum 2 (ranked by A345960).
A011782 counts compositions.
A097805 counts compositions by sum and alternating sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909 (this sequence)/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000346, A008549, A025047, A031443, A031444, A034871, A114121, A238279, A304620, A344651.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]==1&]

A372433 Binary weight (number of ones in binary expansion) of the n-th squarefree number.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 04 2024

Harvey P. Dale, Table of n, a(n) for n = 1..1000
MathOverflow, Are there primes of every Hamming weight?
Wikipedia, Hamming weight.

Restriction of A000120 to A005117.
For prime instead of squarefree we have A014499, zeros A035103.
Counting zeros instead of ones gives A372472, cf. A023416, A372473.
For binary length instead of weight we have A372475.
A003714 lists numbers with no successive binary indices.
A030190 gives binary expansion, reversed A030308.
A048793 lists positions of ones in reversed binary expansion, sum A029931.
A145037 counts ones minus zeros in binary expansion, cf. A031443, A031444, A031448, A097110.
A371571 lists positions of zeros in binary expansion, sum A359359.
A371572 lists positions of ones in binary expansion, sum A230877.
A372515 lists positions of zeros in reversed binary expansion, sum A359400.
A372516 counts ones minus zeros in binary expansion of primes, cf. A177718, A177796, A372538, A372539.
Cf. A039004, A049093, A049094, A059015, A069010, A070939, A073642, A211997, A368494, A372474.

DigitCount[Select[Range[100],SquareFreeQ],2,1]
Total[IntegerDigits[#,2]]&/@Select[Range[200],SquareFreeQ] (* Harvey P. Dale, Feb 14 2025 *)

from math import isqrt
from sympy import mobius
def A372433(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(m).bit_count() # Chai Wah Wu, Aug 02 2024

a(n) = A000120(A005117(n)).

a(n) + A372472(n) = A372475(n) = A070939(A005117(n)).

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

A095073 Primes in whose binary expansion the number of 1-bits is one more than the number of 0-bits.

Original entry on oeis.org

5, 19, 71, 83, 89, 101, 113, 271, 283, 307, 313, 331, 397, 409, 419, 421, 433, 457, 1103, 1117, 1181, 1223, 1229, 1237, 1303, 1307, 1319, 1381, 1427, 1429, 1433, 1481, 1489, 1559, 1579, 1607, 1613, 1619, 1621, 1637, 1699, 1733, 1811, 1861

Offset: 1

Antti Karttunen, Jun 01 2004

			71 is in the sequence because 71_10 = 1000111_2. '1000111' has four 1's and three 0's. - _Indranil Ghosh_, Feb 03 2017

Indranil Ghosh, Table of n, a(n) for n = 1..25000
Antti Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 and A031448. Subset of A095070. Cf. A095053.

Select[Prime[Range[500]], Differences[DigitCount[#, 2]] == {-1} &]

{ forprime(p=2, 2000,
  v=binary(p); s=0;
  for(k=1,#v, s+=if(v[k]==1,+1,-1));
  if(s==1,print1(p,", "))
) }

from sympy import isprime
i=1
j=1
while j<=25000:
    if isprime(i) and bin(i)[2:].count("1")-bin(i)[2:].count("0")==1:
        print(str(j)+" "+str(i))
        j+=1
    i+=1 # Indranil Ghosh, Feb 03 2017

A089648 Numbers whose numbers of zeros and ones in binary representation differ at most by 1.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 9, 10, 12, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86, 88, 89, 90, 92, 97, 98, 99, 100, 101, 102, 104, 105, 106, 108, 112, 113, 114, 116, 120, 135, 139

Offset: 1

Reinhard Zumkeller, Jan 02 2004

A031443 is a subsequence; abs(A037861(a(n))) <= 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Digit Count
Index entries for sequences related to binary expansion of n

Cf. A023416, A000120.

Cf. A037861, union of A031443, A031444 and A031448.

a089648 n = a089648_list !! (n-1)
a089648_list = filter ((<= 1) . abs . a037861) [0..]
-- Reinhard Zumkeller, Mar 31 2015

Select[Range[0,7! ],Abs[DigitCount[ #,2,0]-DigitCount[ #,2,1]]<2 &] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2010 *)

A298847 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the number of ones in the binary expansion of n equals one plus the number of zeros in the binary expansion of a(n).

Original entry on oeis.org

1, 3, 2, 7, 5, 6, 4, 15, 11, 13, 9, 14, 10, 12, 8, 31, 23, 27, 19, 29, 21, 22, 17, 30, 25, 26, 18, 28, 20, 24, 16, 63, 47, 55, 39, 59, 43, 45, 35, 61, 46, 51, 37, 53, 38, 41, 33, 62, 54, 57, 42, 58, 44, 49, 34, 60, 50, 52, 36, 56, 40, 48, 32, 127, 95, 111, 79

Offset: 1

Rémy Sigrist, Jan 27 2018

In other words, for any n > 0, A000120(n) = 1 + A023416(a(n)).
This sequence is a self-inverse permutation of the natural numbers, with fixed points A031448.
We can build an analog of this sequence for any base b > 1:
- let s_b be the sum of digits in base b,
- in particular s_2 = A000120 and s_10 = A007953,
- let l_b be the number of digits in base b,
- in particular l_2 = A070939 and l_10 = A055642,
- let f_b be the lexicographically earliest sequence of distinct positive terms such that, for any n > 0, s_b(n) = 1 + (b-1) * l_b(a(n)) - s_b(a(n)),
- in particular, f_2 = a (this sequence),
- f_b is a self-inverse permutation of the natural numbers,
- l_b(n) = l_b(f_b(n)) for any n > 0,
- f_b(b^k) = b^(k+1) - 1 for any k >= 0,
- see also scatterplots of f_3 and f_10 in Links section.

			The first terms, alongside the binary representations of n and of a(n), are:
  n     a(n)    bin(n)    bin(a(n))
  --    ----    ------    ---------
   1       1         1         1
   2       3        10        11
   3       2        11        10
   4       7       100       111
   5       5       101       101
   6       6       110       110
   7       4       111       100
   8      15      1000      1111
   9      11      1001      1011
  10      13      1010      1101
  11       9      1011      1001
  12      14      1100      1110
  13      10      1101      1010
  14      12      1110      1100
  15       8      1111      1000

Rémy Sigrist, Table of n, a(n) for n = 1..8191
Rémy Sigrist, PARI program for A298847
Rémy Sigrist, Colored scatterplot of the first 2^16 - 1 terms (where the color is function of the Hamming weight of n)
Rémy Sigrist, Scatterplot of the first 3^9 - 1 terms of f_3
Rémy Sigrist, Scatterplot of the first 10^4 - 1 terms of f_10
Index entries for sequences that are permutations of the natural numbers

Cf. A000120, A007953, A023416, A031448, A055642, A070939.

PARI
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A070939(n) = A070939(a(n)) for any n > 0.
a(2^k) = 2^(k+1) - 1 for any k >= 0.
A000120(n) + A000120(a(n)) = 1 + A070939(n) for any n > 0.

A372538 Numbers k such that the number of ones minus the number of zeros in the binary expansion of the k-th prime number is 1.

Original entry on oeis.org

3, 8, 20, 23, 24, 26, 30, 58, 61, 63, 65, 67, 78, 80, 81, 82, 84, 88, 185, 187, 194, 200, 201, 203, 213, 214, 215, 221, 225, 226, 227, 234, 237, 246, 249, 253, 255, 256, 257, 259, 266, 270, 280, 284, 287, 290, 573, 578, 586, 588, 591, 593, 611, 614, 615, 626

Offset: 1

Gus Wiseman, May 13 2024

			The binary expansion of 83 is (1,0,1,0,0,1,1) with ones minus zeros 4 - 3 = 1, and 83 is the 23rd prime, so 23 is in the sequence.
The primes A000040(a(n)) together with their binary expansions and binary indices begin:
     5:           101 ~ {1,3}
    19:         10011 ~ {1,2,5}
    71:       1000111 ~ {1,2,3,7}
    83:       1010011 ~ {1,2,5,7}
    89:       1011001 ~ {1,4,5,7}
   101:       1100101 ~ {1,3,6,7}
   113:       1110001 ~ {1,5,6,7}
   271:     100001111 ~ {1,2,3,4,9}
   283:     100011011 ~ {1,2,4,5,9}
   307:     100110011 ~ {1,2,5,6,9}
   313:     100111001 ~ {1,4,5,6,9}
   331:     101001011 ~ {1,2,4,7,9}
   397:     110001101 ~ {1,3,4,8,9}
   409:     110011001 ~ {1,4,5,8,9}
   419:     110100011 ~ {1,2,6,8,9}
   421:     110100101 ~ {1,3,6,8,9}
   433:     110110001 ~ {1,5,6,8,9}
   457:     111001001 ~ {1,4,7,8,9}
  1103:   10001001111 ~ {1,2,3,4,7,11}
  1117:   10001011101 ~ {1,3,4,5,7,11}
  1181:   10010011101 ~ {1,3,4,5,8,11}
  1223:   10011000111 ~ {1,2,3,7,8,11}

Restriction of A031448 to the primes, positions of ones in A145037.
Taking primes gives A095073, negative A095072.
Positions of ones in A372516, absolute value A177718.
Cf. A066196, A095070-A095075, A177796, A372539.
A000120 counts ones in binary expansion (binary weight), zeros A080791.
A030190 gives binary expansion, reversed A030308.
A035103 counts zeros in binary expansion of primes, firsts A372474.
A048793 lists binary indices, reverse A272020, sum A029931.
A070939 gives the length of an integer's binary expansion.
A101211 lists run-lengths in binary expansion, row-lengths A069010.
A372471 lists binary indices of primes.
Cf. A000040, A003714, A035100, A037861, A053738, A061712, A066195, A104080, A211997, A372429, A372517, A372686.

Select[Range[1000],DigitCount[Prime[#],2,1]-DigitCount[Prime[#],2,0]==1&]

cp's OEIS Frontend

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

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A345910 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum -1.

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A345911 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum 1.

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A345909 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum 1.

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A372433 Binary weight (number of ones in binary expansion) of the n-th squarefree number.

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

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A095073 Primes in whose binary expansion the number of 1-bits is one more than the number of 0-bits.

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A089648 Numbers whose numbers of zeros and ones in binary representation differ at most by 1.

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