A027699 -id:A027699 - cp's OEIS frontend

A095077 Primes with four 1-bits in their binary expansion.

23, 29, 43, 53, 71, 83, 89, 101, 113, 139, 149, 163, 197, 263, 269, 277, 281, 293, 337, 353, 389, 401, 449, 523, 547, 593, 643, 673, 773, 1031, 1049, 1061, 1091, 1093, 1097, 1217, 1283, 1289, 1297, 1409, 1553, 1601, 2069, 2083, 2089, 2129

Offset: 1

Antti Karttunen, Jun 01 2004

T. D. Noe, Table of n, a(n) for n = 1..1000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Subsequence of A027699. First differs from A085448 at n = 19, where a(n)=337, while A085448 continues from there with 311, whose binary expansion has six 1-bits, not four. Cf. A095057.
Cf. A000215 (primes having two bits set), A081091 (three bits set).
Cf. A264908.

Select[Prime[Range[320]], Plus@@IntegerDigits[#, 2] == 4 &] (* Alonso del Arte, Jan 11 2011 *)
Select[ Flatten[ Table[2^i + 2^j + 2^k + 1, {i, 3, 11}, {j, 2, i - 1}, {k, j - 1}]], PrimeQ] (* Robert G. Wilson v, Jul 30 2016 *)

bits1_4(x) = { nB = floor(log(x)/log(2)); z = 0;
for(i=0,nB,if(bittest(x,i),z++;if(z>4,return(0););););
if(z == 4, return(1);, return(0););};
forprime(x=17,2129,if(bits1_4(x),print1(x, ", ");););
\\ Washington Bomfim, Jan 11 2011

is(n)=isprime(n) && hammingweight(n)==4 \\ Charles R Greathouse IV, Jul 30 2016

list(lim)=my(v=List(),t); for(a=3,logint(lim\=1,2), for(b=2,a-1, for(c=1,b-1, t=1<lim, return(Vec(v))); if(isprime(t), listput(v,t))))); Vec(v) \\ Charles R Greathouse IV, Jul 30 2016

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A095077_gen(): # generator of terms
    return filter(isprime,map(lambda s:int('1'+''.join(s)+'1',2),(s for l in count(2) for s in multiset_permutations('0'*(l-2)+'11'))))
A095077_list = list(islice(A095077_gen(),30)) # Chai Wah Wu, Jul 19 2022

A130911 a(n) is the number of primes with odd binary weight among the first n primes minus the number with an even binary weight.

Original entry on oeis.org

1, 0, -1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 3, 2, 3, 2, 3, 4, 5, 4, 5, 6, 5, 4, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 7, 8, 9, 8, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 21, 20, 19, 20, 19, 18, 19, 18, 19, 18, 19, 18, 19, 18, 17, 16, 15, 14, 15, 14, 15, 14, 13, 14, 13, 14, 15, 16, 17, 18, 19, 20, 19, 20, 19, 20, 19, 18, 19, 20, 21, 20, 19

Offset: 1

T. D. Noe, Jun 08 2007

Prime race between evil primes (A027699) and odious primes (A027697).
Shevelev conjectures that a(n) >= 0 for n > 3. Surprisingly, the conjecture also appears to be true if we count zeros instead of ones in the binary representation of prime numbers.
The conjecture is true for primes up to at least 10^13. Mauduit and Rivat prove that half of all primes are evil. - T. D. Noe, Feb 09 2009

T. D. Noe, Table of n, a(n) for n = 1..10000
CNRS Press release, The sum of digits of prime numbers is evenly distributed, May 12, 2010.
Christian Mauduit and Joël Rivat, Sur un problème de Gelfond: la somme des chiffres des nombres premiers, Annals Math., 171 (2010), 1591-1646.
ScienceDaily, Sum of Digits of Prime Numbers Is Evenly Distributed: New Mathematical Proof of Hypothesis, May 12, 2010.
Vladimir Shevelev, A conjecture on primes and a step towards justification, arXiv:0706.0786 [math.NT], 2007.
Vladimir Shevelev, On excess of odious primes, arXiv:0707.1761 [math.NT], 2007.

Cf. A095005, A095006.
Cf. A199399, A027697, A027698, A027699, A027700, A200244, A200245, A200246, A200247.
Cf. A156549 (race between primes having an odd/even number of zeros in binary).

cnt=0; Table[p=Prime[n]; If[EvenQ[Count[IntegerDigits[p,2],1]], cnt--, cnt++ ]; cnt, {n,10000}]
Accumulate[If[OddQ[DigitCount[#,2,1]],1,-1]&/@Prime[Range[100]]] (* Harvey P. Dale, Aug 09 2013 *)

f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==1,s++)); return(s%2)};nO=0;nE=0;forprime(p=2,520,if(f(p),nO++, nE++);an=nO-nE;print1(an,", ")) \\ Washington Bomfim, Jan 14 2011

from sympy import nextprime
from itertools import islice
def agen():
    p, evod = 2, [0, 1]
    while True:
        yield evod[1] - evod[0]
        p = nextprime(p); evod[bin(p).count('1')%2] += 1
print(list(islice(agen(), 97))) # Michael S. Branicky, Dec 21 2021

a(n) = (number of odious primes <= prime(n)) - (number of evil primes <= prime(n)).

a(n) = A200247(n) - A200246(n).

Edited by N. J. A. Sloane, Nov 16 2011

A066148 Primes with an even number of 0's in binary expansion.

Original entry on oeis.org

3, 7, 19, 31, 43, 53, 67, 73, 79, 97, 103, 107, 109, 127, 139, 149, 163, 197, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443, 457, 463, 487, 491, 499, 523, 547, 571, 593, 599, 619, 643, 673, 683, 691, 739, 751, 773, 797, 811, 821

Offset: 1

R. K. Guy, Dec 13 2001

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A066149, A027699, A027697.

Cf. A059010

Select[Prime[Range[200]],EvenQ[DigitCount[#,2,0]]&] (* Harvey P. Dale, Mar 04 2017 *)

a066148(m) = local(p,v,z); forprime(p=2,m,v=binary(p); z=0; for(j=1,matsize(v)[2], if(v[j]==0,z++)); if(z%2==0,print1(p,",")))
a066148(850)

f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==0, s++));return(1-s%2)};forprime(p=3,821,if(f(p),print1(p,", "))) \\ Washington Bomfim, Jan 14 2011

forprime(p=2, 10^3, if( #select(x->x==0, digits(p,2))%2==0, print1(p,", "))); \\ Joerg Arndt, Jun 16 2018

More terms from Vladeta Jovovic and Klaus Brockhaus, Dec 13 2001

A066149 Primes with an odd number of 0's in binary expansion.

Original entry on oeis.org

2, 5, 11, 13, 17, 23, 29, 37, 41, 47, 59, 61, 71, 83, 89, 101, 113, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 277, 281, 293, 311, 317, 337, 347, 349, 353, 359, 373, 383, 389, 401, 449, 461

Offset: 1

R. K. Guy, Dec 13 2001

			17 is in the sequence because 17 is a prime and 17 = 10001_2. '10001' has three 0's. - _Indranil Ghosh_, Feb 06 2017

Indranil Ghosh, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe)

Cf. A066148, A027699, A027697.

Cf. A059009.

Select[ Prime[ Range[ PrimePi[ 1000 ] ] ], OddQ[ Count[ IntegerDigits[ #, 2 ], 0 ] ]& ]

forprime(p=2, 10^3, if( #select(x->x==0, digits(p, 2))%2==1, print1(p, ", "))); \\ Joerg Arndt, Jun 16 2018

More terms from Vladeta Jovovic and Klaus Brockhaus, Dec 13 2001

A027698 Numbers k such that the k-th prime has an odd number of 1's in its binary expansion.

Original entry on oeis.org

1, 4, 5, 6, 8, 11, 12, 13, 15, 17, 18, 19, 21, 22, 25, 27, 28, 29, 31, 32, 33, 36, 37, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 58, 61, 63, 65, 67, 73, 75, 78, 80, 81, 82, 83, 84, 85, 86, 88, 90, 93, 94, 95, 98, 100, 102, 103, 104, 106, 107, 110, 111, 112

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A027697, A027699, A027700.

Select[ Range[ 150 ], OddQ[ Length[ Cases[ IntegerDigits[ Prime[ # ], 2 ], 1 ] ] ]& ]
Select[Range[200],OddQ[DigitCount[Prime[#],2,1]]&] (* Harvey P. Dale, Sep 19 2021 *)

More terms from Erich Friedman

A027700 Numbers k such that the k-th prime has an even number of 1's in its binary expansion.

Original entry on oeis.org

2, 3, 7, 9, 10, 14, 16, 20, 23, 24, 26, 30, 34, 35, 38, 45, 55, 56, 57, 59, 60, 62, 64, 66, 68, 69, 70, 71, 72, 74, 76, 77, 79, 87, 89, 91, 92, 96, 97, 99, 101, 105, 108, 109, 114, 117, 122, 124, 125, 131, 133, 137, 139, 141, 142, 146, 147, 148, 150, 152, 154, 155, 159, 165, 166, 170, 173, 176, 178

Offset: 1

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A027697, A027698, A027699.

Position[Table[If[EvenQ[DigitCount[n,2,1]],1,0],{n,Prime[Range[ 200]]}],1]//Flatten (* Harvey P. Dale, Jul 11 2017 *)

n=0;forprime(p=2,97,n++;if(hammingweight(p)%2==0,print1(n", "))) \\ Charles R Greathouse IV, Sep 24 2012

Extended (and corrected) by Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A095282 Primes whose binary-expansion ends with an even number of 1's.

Original entry on oeis.org

2, 3, 11, 19, 43, 47, 59, 67, 79, 83, 107, 131, 139, 163, 179, 191, 211, 227, 239, 251, 271, 283, 307, 331, 347, 367, 379, 419, 431, 443, 463, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 719, 739, 751, 787, 811, 827, 859

Offset: 1

Antti Karttunen, Jun 04 2004

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
A. Karttunen and J. Moyer, C-program for computing the initial terms of this sequence

Intersection of A000040 & (complement of A079523). Complement of A095283 in A000040. Cf. A027699, A095292.

q:= proc(n) local i, l, r; l, r:= convert(n, base, 2), 0;
      for i to nops(l) while l[i]=1 do r:=r+1 od; is(r, even)
    end:
select(q, [ithprime(i)$i=1..200])[];  # Alois P. Heinz, Dec 15 2019

been1Q[n_]:=Module[{c=Split[IntegerDigits[n,2]][[-1]]},c[[1]]==1&&EvenQ[ Length[ c]]]; Join[{2},Select[Prime[Range[150]],been1Q]] (* Harvey P. Dale, Dec 14 2019 *)

is(n)=valuation(n+1,2)%2==0 && isprime(n) \\ Charles R Greathouse IV, Oct 09 2013

A268476 Balanced evil primes: primes with an even number of runs of 1's in their binary expansion.

Original entry on oeis.org

5, 11, 13, 17, 19, 23, 29, 47, 59, 61, 67, 71, 79, 97, 103, 113, 131, 149, 173, 181, 191, 193, 199, 223, 227, 239, 241, 251, 257, 263, 271, 277, 293, 331, 337, 347, 349, 373, 383, 421, 449, 463, 479, 487, 499, 503, 509, 557, 587, 593, 599, 601, 613, 617, 619

Offset: 1

Vladimir Shevelev, Feb 05 2016

Primes in A268412. Complement of A268477.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Cf. A027699, A268412, A268477.

Select[Prime@ Range@ 120, EvenQ@ Length[Split@ IntegerDigits[#, 2] /. {0, _} -> Nothing] &] (* Michael De Vlieger, Feb 08 2016 *)

from sympy import prime
A268476_list = [p for p in (prime(i) for i in range(1,10**6)) if not len(list(filter(bool,format(p,'b').split('0')))) % 2] # Chai Wah Wu, Mar 01 2016

More terms from Peter J. C. Moses, Feb 05 2016

A268477 Balanced odious primes: primes with an odd number of runs of 1's in their binary expansion.

Original entry on oeis.org

2, 3, 7, 31, 37, 41, 43, 53, 73, 83, 89, 101, 107, 109, 127, 137, 139, 151, 157, 163, 167, 179, 197, 211, 229, 233, 269, 281, 283, 307, 311, 313, 317, 353, 359, 367, 379, 389, 397, 401, 409, 419, 431, 433, 439, 443, 457, 461, 467, 491, 521, 523, 541, 547, 563

Offset: 1

Vladimir Shevelev, Feb 05 2016

Primes from A268415.
According to our 2007-conjecture, if pi_1(m) is the number of evil primes (A027699) not exceeding m and pi_2(m) is the number of odious primes (A027697) not exceeding m, then pi_1(m)<=pi_2(m) for all natural m except m=5 and m=6.
In the "balance" case of A268476,A268477, most likely, none of two types of primes
is in the majority beginning with any place.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv:1603.04434 [math.NT], 2016.

Cf. A027697, A268415, A268476.

Select[Prime@ Range@ 108, OddQ@ Length[Split@ IntegerDigits[#, 2] /. {0, _} -> Nothing] &] (* Michael De Vlieger, Feb 08 2016 *)

from sympy import prime
A268477_list = [p for p in (prime(i) for i in range(1,10**6)) if len(list(filter(bool,format(p,'b').split('0')))) % 2] # Chai Wah Wu, Mar 01 2016

More terms from Peter J. C. Moses, Feb 05 2016

A357761 a(n) = A227872(n) - A356018(n).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 2, 4, -1, 0, 2, 0, 2, 4, -2, 5, 0, -2, 2, 0, 2, 4, 0, 0, 1, 4, -2, 6, 0, -4, 2, 6, 0, 0, 2, -3, 2, 4, 0, 0, 2, 4, 0, 6, -4, 0, 2, 0, 3, 2, -2, 6, 0, -4, 2, 8, 0, 0, 2, -6, 2, 4, 0, 7, 0, 0, 2, 0, 0, 4, 0, -4, 2, 4, -2, 6, 2, 0, 2, 0, -1, 4, 0

Offset: 1

Amiram Eldar, Oct 12 2022

The excess of the number of odious (A000069) divisors of n over the number of evil (A001969) divisors of n.

Every integer occurs in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000069, A000290 (positions of odd terms), A001969, A027697, A027699, A106400, A227872, A230851 (positions of 0's), A356018, A357762.

Similar sequences: A046660, A048272.

a[n_] := -DivisorSum[n, (-1)^DigitCount[#, 2, 1] &]; Array[a, 100]

a(n) = -sumdiv(n, d, (-1)^hammingweight(d));

a(n) = -Sum_{d|n} A106400(d).
a(n) = A000005(n) - 2*A356018(n).
a(n) = 2*A227872(n) - A000005(n).
a(n) = 0 iff n is in A230851.
a(n) == 1 (mod 2) iff n is a square (A000290).
a(2^n) = n + 1.
a(p*2^n) = 0 when p is an evil prime (A027699).
a(p^2*2^n) = n + 1 when p is an evil prime (A027699) and p^2 is odious, and when p is an odd odious prime (A027697) and p^2 is evil.
a(p^2*2^n) = -(n+1) when p is an evil prime and p^2 is also evil.
a(p^2*2^n) = 3*(n+1) when p is an odd odious prime and p^2 is also odious.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -Sum_{k>=1} A106400(k)/k = 1.196283264... (A357762).

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A095077 Primes with four 1-bits in their binary expansion.

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A130911 a(n) is the number of primes with odd binary weight among the first n primes minus the number with an even binary weight.

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A066148 Primes with an even number of 0's in binary expansion.

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A066149 Primes with an odd number of 0's in binary expansion.

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A027698 Numbers k such that the k-th prime has an odd number of 1's in its binary expansion.

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A027700 Numbers k such that the k-th prime has an even number of 1's in its binary expansion.

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A095282 Primes whose binary-expansion ends with an even number of 1's.

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A268476 Balanced evil primes: primes with an even number of runs of 1's in their binary expansion.

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