A027697 -id:A027697 - cp's OEIS frontend

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.

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

A235479 Primes whose base-2 representation also is the base-9 representation of a prime.

Original entry on oeis.org

11, 13, 19, 41, 79, 109, 137, 151, 167, 191, 193, 199, 227, 239, 271, 307, 313, 421, 431, 433, 457, 487, 491, 521, 563, 613, 617, 659, 677, 709, 727, 757, 929, 947, 1009, 1033, 1051, 1249, 1483, 1693, 1697, 1709, 1721, 1831, 1951, 1979, 1987, 1993

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
A subsequence of A027697, A050150, A062090 and A176620.

			11 = 1011_2 and 1011_9 = 6571 are both prime, so 11 is a term.

Robert Price, Table of n, a(n) for n = 1..1958
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235466 ⊂ A077723, A235266, A152079, A235475 - A235478, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

is(p,b=9)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

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)

A095283 Primes whose binary-expansion ends with an odd number of 1's.

Original entry on oeis.org

5, 7, 13, 17, 23, 29, 31, 37, 41, 53, 61, 71, 73, 89, 97, 101, 103, 109, 113, 127, 137, 149, 151, 157, 167, 173, 181, 193, 197, 199, 223, 229, 233, 241, 257, 263, 269, 277, 281, 293, 311, 313, 317, 337, 349, 353, 359, 373, 383, 389, 397, 401, 409

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 & A079523. Complement of A095282 in A000040. Cf. A027697, A095293.

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, odd)
    end:
select(q, [ithprime(i)$i=1..150])[];  # Alois P. Heinz, Dec 15 2019

Select[Prime[Range[100]], MatchQ[IntegerDigits[#, 2], {b:(1)..}|{_, 0, b:(1)..} /; OddQ[Length[{b}]]]&] (* Jean-François Alcover, Jan 03 2022 *)

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

from sympy import isprime
def ok(n): b = bin(n); return (len(b)-len(b.rstrip("1")))%2 and isprime(n)
print([k for k in range(1, 401) if ok(k)]) # Michael S. Branicky, Jan 03 2022

A235477 Primes whose base-2 representation also is the base-7 representation of a prime.

Original entry on oeis.org

2, 31, 47, 59, 103, 107, 173, 179, 181, 199, 211, 227, 229, 233, 367, 409, 443, 463, 487, 701, 743, 757, 823, 827, 877, 911, 919, 967, 1009, 1123, 1163, 1291, 1321, 1367, 1373, 1447, 1493, 1571, 1583, 1597, 1609, 1627, 1657, 1669, 1721, 1831, 1933, 1987

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
A subsequence of A027697, A015919, A197636 (conjectural).

			31 = 11111_2 and 11111_7 = 2801 are both prime, so 31 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235464 ⊂ A077721, A235475, A152079, A235266, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,2],7]]&] (* Harvey P. Dale, May 08 2021 *)

is(p,b=7)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

A238186 Primes with odd Hamming weight that as polynomials over GF(2) are reducible.

Original entry on oeis.org

79, 107, 127, 151, 173, 179, 181, 199, 223, 227, 233, 251, 271, 307, 331, 367, 409, 421, 431, 439, 443, 457, 491, 521, 541, 569, 577, 641, 653, 659, 709, 727, 733, 743, 809, 823, 829, 919, 941, 947, 991, 997, 1009, 1021, 1087, 1109, 1129, 1171, 1187, 1201, 1213, 1231, 1249, 1259, 1301, 1321, 1327, 1361, 1373

Offset: 1

Joerg Arndt, Feb 19 2014

nonn

Subsequence of A091209 (see comments there).

			79 is a term because 79 = 1001111_2 which corresponds to the polynomial x^6 + x^3 + x^2 + x + 1, but over GF(2) we have x^6 + x^3 + x^2 + x + 1 = (x^2 + x + 1)*(x^4 + x^3 + 1). - _Jianing Song_, May 10 2021

Intersection of A000069 and A091209.
Intersection of A027697 and A091242.
Equals the set difference of A027697 and A091206.

forprime(p=2, 10^4, if( (hammingweight(p)%2==1) && ! polisirreducible( Mod(1,2)*Pol(binary(p)) ), print1(p,", ") ) );

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

cp's OEIS Frontend

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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A235479 Primes whose base-2 representation also is the base-9 representation of a prime.

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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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A095283 Primes whose binary-expansion ends with an odd number of 1's.

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A235477 Primes whose base-2 representation also is the base-7 representation of a prime.

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