A027699 -id:A027699 - cp's OEIS frontend

A095006 Number of evil primes (A027699) in range ]2^n,2^(n+1)].

1, 1, 0, 3, 2, 5, 4, 23, 27, 62, 95, 222, 367, 777, 1269, 2910, 4859, 10140, 17714, 36714, 66020, 133400, 245959, 493532, 916913, 1822087, 3428633, 6782008, 12870735, 25339113, 48419194, 95194890, 182818705, 358637144, 691891351, 1355985684, 2625053871, 5142673207

Offset: 1

Antti Karttunen and Labos Elemer, Jun 01 2004

nonn

			From _Michael De Vlieger_, Feb 27 2017: (Start)
a(2) = 1 since between 2^2 and 2^3 only the prime 5 (binary 11) has an even number of 1s.
a(3) = 0 since none of the primes between 2^3 and 2^4 have an even number of 1s in their binary expansions.
a(4) = 3 since the primes 17, 23, and 29 have an even number of 1s in their binary expansions (i.e., 10001, 10111, 11101). (End)

Antti Karttunen and John Moyer, C-program for computing the initial terms of this sequence.
Index entries for sequences related to occurrences of various subsets of primes in range ]2^n,2^(n+1)].

Cf. A027699, A036378, A095005.

Table[m = Count[Prime@ Range[PrimePi[2^n] + 1, PrimePi[2^(n + 1) - 1]], k_ /; EvenQ@ DigitCount[k, 2, 1]]; Print@ m; m, {n, 24}] (* Michael De Vlieger, Feb 27 2017 *)

a(n) = A036378(n) - A095005(n).

a(34)-a(38) from Amiram Eldar, Jun 20 2024

A177748 First primes of record chains of consecutive primes such that all of them are evil (A027699).

Original entry on oeis.org

3, 257, 337, 4423, 4919, 30431, 66841, 514271, 14490383, 231234569, 325923613, 640085473, 1600993259, 7180164577, 8069913503, 86933359951, 284331217637, 1128352801153, 1209935587291, 2454267258251, 2945783287813

Offset: 1

Vladimir Shevelev, Dec 12 2010

The first lengths of such chains of primes are: 2, 3, 5, 7, 8, 12, 16, ..., cf. A177801.

			257 is the first evil prime followed by two consecutive primes (263,269) which are also evil. Thus it is record length 3.

Cf. A001969, A000069, A027697, A027699, A177798 (odious version), A177801.

{l=0;sp=0;r=0; forprime( p=1, default(primelimit), if( norml2(binary(p))%2, l>r & !print1(sp", ") & r=l; l & l=0, l || sp=p; l++))} \\ M. F. Hasler, Dec 12 2010

More terms from D. S. McNeil, Dec 12 2010

a(18)-a(21) from Amiram Eldar, Dec 09 2020

A230353 Products of 3 evil primes (A027699) p,q,r, such that numbers pq, pr, qr, and pq*r are odious (A000069).

Original entry on oeis.org

575, 1775, 2075, 2225, 2825, 3475, 6575, 8381, 8675, 8825, 8975, 8993, 10235, 11225, 11675, 11975, 12035, 12167, 12905, 13075, 14275, 14825, 18745, 19925, 21575, 22881, 23943, 24389, 25325, 25775, 26765, 27575, 30189, 30925, 30981, 31433, 32223, 32675, 32975

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Oct 16 2013

These numbers are products of 3 evil numbers (A001969) but not represented as products of two evil numbers (A230213).

			For triple of evil primes {3,29,263} numbers 3*29 = 87, 3*263 = 789, 29*263 = 7627 and 3*29*263 = 22881. Thus 22881 is in the sequence.

Charles R Greathouse IV and Peter J. C. Moses, Table of n, a(n) for n = 1..10000 (first 500 from Moses)

Cf. A000069, A001969, A027699, A230213.

od(n)=hammingweight(n)%2
list(lim)=my(v=List(),pq); forprime(p=23,lim\25, if(od(p), next); forprime(q=5,min(lim\(3*p),p), if(od(q) || !od(pq=p*q), next); forprime(r=3,min(lim\pq,q), if(!od(r) && od(q*r) && od(p*r) && od(pq*r), listput(v, pq*r))))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Nov 01 2013

A231254 Odious primes p (A027697) such that p^2 and p^3 are evil (A027699).

Original entry on oeis.org

37, 47, 107, 137, 233, 331, 463, 491, 557, 587, 607, 631, 653, 733, 823, 829, 883, 947, 971, 997, 1153, 1187, 1193, 1231, 1249, 1321, 1327, 1493, 1543, 1567, 1663, 1667, 1669, 1709, 1787, 1801, 1933, 1987, 2011, 2027, 2087, 2143, 2161, 2213, 2269, 2273, 2311

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Nov 06 2013

Sequence {a(n)^4} is a subsequence of A227891 such that a(1)^4 = 1874161 is the smallest power of an odious prime that is in A227891.

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

Cf. A001969, A027697, A027699, A227891.

evilQ[n_]:=EvenQ[DigitCount[n,2][[1]]];
odiousQ[n_]:=OddQ[DigitCount[n, 2][[1]]];
Select[Range[2000],PrimeQ[#]&&odiousQ[#]&&evilQ[#^2]&&evilQ[#^3]&] (* Peter J. C. Moses, Nov 08 2013 *)

A360648 Fully multiplicative with a(A027697(k)) = A027699(k) and a(A027699(k)) = A027697(k) for any k > 0.

Original entry on oeis.org

1, 3, 2, 9, 7, 6, 5, 27, 4, 21, 17, 18, 23, 15, 14, 81, 11, 12, 29, 63, 10, 51, 13, 54, 49, 69, 8, 45, 19, 42, 43, 243, 34, 33, 35, 36, 53, 87, 46, 189, 71, 30, 31, 153, 28, 39, 83, 162, 25, 147, 22, 207, 37, 24, 119, 135, 58, 57, 89, 126, 101, 129, 20, 729

Offset: 1

Rémy Sigrist, Feb 15 2023

In other words, we replace odious prime numbers with evil prime numbers and vice versa.

This sequence is a self-inverse permutation of the positive integers.

			For n = 84:
- 82 = 2^2 * 3 * 7,
- a(2) = a(A027697(1)) = A027699(1) = 3,
- a(3) = a(A027699(1)) = A027697(1) = 2,
- a(7) = a(A027697(2)) = A027699(2) = 5,
- so a(84) = 3^2 * 2 * 5 = 90.

Rémy Sigrist, Scatterplot of (n, a(n)) such that n, a(n) <= 100000
Rémy Sigrist, PARI program
Index entries for sequences that are permutations of the natural numbers

Cf. A061898, A027697, A027699, A341090.

PARI
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See Links section.
```

A131122 Even numbers that are not the sum of an evil prime (A027699) and an odious prime (A027697).

Original entry on oeis.org

2, 4, 6, 8, 20, 26, 32, 38, 68, 86, 92, 98, 128, 164, 188, 278, 302, 512, 2048, 8192, 32768, 131072, 524288, 2097152, 8388608, 33554432, 134217728, 536870912

Offset: 1

T. D. Noe, Jun 15 2007

Every power of 2 with an odd exponent is a term.

Every pair of primes that sum to 4^k, with k > 1, consists of an evil prime and an odious prime. The smallest example is 16 = 3+13 = 5+11; 3 and 5 are evil, 11 and 13 are odious.

			32 is here because 32 = 3+29 = 13+19, 3 and 29 are both odious, and 13 and 19 are both evil.

Cf. A004171 (odd powers of 2), A027697, A027699.

isok(n) = {if ((n % 2) == 0, forprime(p=3, n, if ((norml2(binary(p))%2==1) && (isprime(q=n-p)) && (!bittest(norml2(binary(q)), 0)), return (0));); return (1);); return(0);}
lista(nn) = forstep(n=2, nn, 2, if (isok(n), print1(n, ", "))); \\ Michel Marcus, Oct 14 2018

a(23)-a(28) from Michel Marcus, Oct 14 2018

A001969 Evil numbers: nonnegative integers with an even number of 1's in their binary expansion.

Original entry on oeis.org

0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58, 60, 63, 65, 66, 68, 71, 72, 75, 77, 78, 80, 83, 85, 86, 89, 90, 92, 95, 96, 99, 101, 102, 105, 106, 108, 111, 113, 114, 116, 119, 120, 123, 125, 126, 129

Offset: 1

N. J. A. Sloane

This sequence and A000069 give the unique solution to the problem of splitting the nonnegative integers into two classes in such a way that sums of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000028, A000379.
In French: les nombres païens.
Theorem: First differences give A036585. (Observed by Franklin T. Adams-Watters.)
Proof from Max Alekseyev, Aug 30 2006 (edited by N. J. A. Sloane, Jan 05 2021): (Start)
Observe that if the last bit of a(n) is deleted, we get the nonnegative numbers 0, 1, 2, 3, ... in order.
The last bit in a(n+1) is 1 iff the number of bits in n is odd, that is, iff A010060(n+1) is 1.
So, taking into account the different offsets here and in A010060, we have a(n) = 2*(n-1) + A010060(n-1).
Therefore the first differences of the present sequence equal 2 + first differences of A010060, which equals A036585.  QED (End)
Integers k such that A010060(k-1)=0. - Benoit Cloitre, Nov 15 2003
Indices of zeros in the Thue-Morse sequence A010060 shifted by 1. - Tanya Khovanova, Feb 13 2009
Conjecture, checked up to 10^6: a(n) is also the sequence of numbers k representable as k = ror(x) XOR rol(x) (for some integer x) where ror(x)=A038572(x) is x rotated one binary place to the right, rol(x)=A006257(x) is x rotated one binary place to the left, and XOR is the binary exclusive-or operator. - Alex Ratushnyak, May 14 2016
From Charlie Neder, Oct 07 2018: (Start)
Conjecture is true: ror(x) and rol(x) have an even number of 1 bits in total (= 2 * A000120(x)), and XOR preserves the parity of this total, so the resulting number must have an even number of 1 bits. An x can be constructed corresponding to a(n) like so:
If the number of bits in a(n) is even, add a leading 0 so a(n) is 2k+1 bits long.
Do an inverse shuffle on a(n), then "divide" by 11, rotate the result k bits to the right, and shuffle to get x. (End)
Numbers of the form m XOR (2*m) for some m >= 0. - Rémy Sigrist, Feb 07 2021
The terms "evil numbers" and "odious numbers" were coined by Richard K. Guy, c. 1976 (Haque and Shallit, 2016) and appeared in the book by Berlekamp et al. (Vol. 1, 1st ed., 1982). - Amiram Eldar, Jun 08 2021

Elwyn R. Berlekamp, John H. Conway, Richard K. Guy, Winning Ways for Your Mathematical Plays, Volume 1, 2nd ed., A K Peters, 2001, chapter 14, p. 110.
Hugh L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
Donald J. Newman, A Problem Seminar, Springer; see Problem #89.
Vladimir S. Shevelev, On some identities connected with the partition of the positive integers with respect to the Morse sequence, Izv. Vuzov of the North-Caucasus region, Nature sciences 4 (1997), 21-23 (Russian).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Jean-Paul Allouche and Henri Cohen, Dirichlet series and curious infinite products, Bull. London Math. Soc., Vol. 17 (1985), pp. 531-538.
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, Theoretical Computer Sci., Vol. 98 (1992), pp. 163-197; DOI.
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214 [math.NT], 2014.
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, Aequationes mathematicae, Vol. 90 (2016), pp. 341-353; alternative link.
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 19.
Chris Bernhardt, Evil twins alternate with odious twins, Math. Mag., Vol. 82, No. 1 (2009), pp. 57-62; alternative link.
Joshua N. Cooper, Dennis Eichhorn and Kevin O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, Vol. 2, No. 4 (2006), pp. 499-522; arXiv preprint, arXiv:math/0506496 [math.NT], 2005.
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math., Vol. 312, No. 1 (2012), pp. 42-46.
E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, (French) [Sums of digits and almost primes] Math. Ann., Vol. 305, No. 3 (1996), pp. 571-599. MR1397437 (97k:11029)
Sajed Haque, Chapter 3.2 of Discriminators of Integer Sequences, thesis, University of Waterloo, Ontario, Canada, 2017. See p. 38.
Sajed Haque and Jeffrey Shallit, Discriminators and k-Regular Sequences Integers, Vol. 16 (2016), Article A76; arXiv preprint, arXiv:1605.00092 [cs.DM], 2016.
Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.
J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull., Vol. 2, No. 2 (1959), pp. 85-89.
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., Vol. 3, No. 4 (1974), pp. 255-261.
Jeffrey O. Shallit, On infinite products associated with sums of digits, J. Number Theory, Vol. 21, No. 2 (1985), pp. 128-134.
Jeffrey Shallit, Frobenius Numbers and Automatic Sequences, arXiv:2103.10904 [math.NT], 2021.
Jeffrey Shallit, Additive Number Theory via Automata and Logic, arXiv:2112.13627 [math.NT], 2021.
Vladimir Shevelev and Peter J. C. Moses, Tangent power sums and their applications, arXiv preprint arXiv:1207.0404 [math.NT], 2012. - From _N. J. A. Sloane_, Dec 17 2012
Vladimir Shevelev and Peter J. C. Moses, A family of digit functions with large periods, arXiv preprint arXiv:1209.5705 [math.NT], 2012.
Vladimir Shevelev and Peter J. C. Moses, Tangent power sums and their applications, INTEGERS, Vol. 14 (2014) #64.
Eric Weisstein's World of Mathematics, Evil Number.
Index entries for sequences related to binary expansion of n
Index entries for "core" sequences

Complement of A000069 (the odious numbers). Cf. A133009.
a(n)=2*n+A010060(n)=A000069(n)-(-1)^A010060(n). Cf. A018900.
The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015.
Cf. A036585 (differences), A010060, A006364.
For primes see A027699, also A130593.
Cf. A006068, A048724, A059010, A094677.

a001969 n = a001969_list !! (n-1)
a001969_list = [x | x <- [0..], even $ a000120 x]
-- Reinhard Zumkeller, Feb 01 2012

[ n : n in [0..129] | IsEven(&+Intseq(n,2)) ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

s := proc(n) local i,j,ans; ans := [ ]; j := 0; for i from 0 while jA001969 := n->t1[n]; # s(k) gives first k terms.
# Alternative:
seq(`if`(add(k, k=convert(n,base,2))::even, n, NULL), n=0..129); # Peter Luschny, Jan 15 2021
# alternative for use outside this sequence
isA001969 := proc(n)
    add(d,d=convert(n,base,2)) ;
    type(%,'even') ;
end proc:
A001969 := proc(n)
    option remember ;
    local a;
    if n = 0 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA001969(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A001969(n),n=1..200) ; # R. J. Mathar, Aug 07 2022

Select[Range[0,300], EvenQ[DigitCount[ #, 2][[1]]] &]
a[ n_] := If[ n < 1, 0, With[{m = n - 1}, 2 m + Mod[-Total@IntegerDigits[m, 2], 2]]]; (* Michael Somos, Jun 09 2019 *)

a(n)=n-=1; 2*n+subst(Pol(binary(n)),x,1)%2

a(n)=if(n<1,0,if(n%2==0,a(n/2)+n,-a((n-1)/2)+3*n))

a(n)=2*(n-1)+hammingweight(n-1)%2 \\ Charles R Greathouse IV, Mar 22 2013

def ok(n): return bin(n)[2:].count('1') % 2 == 0
print(list(filter(ok, range(130)))) # Michael S. Branicky, Jun 02 2021

from itertools import chain, count, islice
def A001969_gen(): # generator of terms
    return chain((0,),chain.from_iterable((sorted(n^ n<<1 for n in range(2**l,2**(l+1))) for l in count(0))))
A001969_list = list(islice(A001969_gen(),30)) # Chai Wah Wu, Jun 29 2022

def A001969(n): return ((m:=n-1).bit_count()&1)+(m<<1) # Chai Wah Wu, Mar 03 2023

a(n+1) - A001285(n) = 2n-1 has been verified for n <= 400. - John W. Layman, May 16 2003 [This can be directly verified by comparing Ralf Stephan's formulas for this sequence (see below) and for A001285. - Jianing Song, Nov 04 2024]
Note that 2n+1 is in the sequence iff 2n is not and so this sequence has asymptotic density 1/2. - Franklin T. Adams-Watters, Aug 23 2006
a(n) = (1/2) * (4n - 3 - (-1)^A000120(n-1)). - Ralf Stephan, Sep 14 2003
G.f.: Sum_{k>=0} (t(3+2t+3t^2)/(1-t^2)^2) * Product_{l=0..k-1} (1-x^(2^l)), where t = x^2^k. - Ralf Stephan, Mar 25 2004
a(2*n+1) + a(2*n) = A017101(n-1) = 8*n-5.
a(2*n) - a(2*n-1) gives the Thue-Morse sequence (3, 1 version): 3, 1, 1, 3, 1, 3, 3, 1, 1, 3, .... A001969(n) + A000069(n) = A016813(n-1) = 4*n-3. - Philippe Deléham, Feb 04 2004
a(1) = 0; for n > 1: a(n) = 3*n-3 - a(n/2) if n even, a(n) = a((n+1)/2)+n-1 if n odd.
Let b(n) = 1 if sum of digits of n is even, -1 if it is odd; then Shallit (1985) showed that Product_{n>=0} ((2n+1)/(2n+2))^b(n) = 1/sqrt(2).
a(n) = 2n - 2 + A010060(n-1). - Franklin T. Adams-Watters, Aug 28 2006
A005590(a(n-1)) <= 0. - Reinhard Zumkeller, Apr 11 2012
A106400(a(n-1)) = 1. - Reinhard Zumkeller, Apr 29 2012
a(n) = (a(n-1) + 2) XOR A010060(a(n-1) + 2). - Falk Hüffner, Jan 21 2022
a(n+1) = A006068(n) XOR (2*A006068(n)). - Rémy Sigrist, Apr 14 2022

More terms from Robin Trew (trew(AT)hcs.harvard.edu)

A014499 Number of 1's in binary representation of n-th prime.

Original entry on oeis.org

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

Offset: 1

Ingemar Assarsjo (ingemar(AT)binomen.se)

a(n) is the rank of prime(n) in the base-2 dominance order on the natural numbers. - Tom Edgar, Mar 25 2014

			From _M. F. Hasler_, Mar 03 2023: (Start)
a(n) = 1 only for p(n = 1) = 2, the only prime equal to a power of 2.
a(n) = 2 for n in A159611 = A000720(A019434) = {2, 3, 7, 55, 6543} (probably complete), the Fermat primes F[k] = 2^2^k + 1 with k = 0, 1, 2, 3, 4. (On the graph one can distinctly see a(6543) = 2 corresponding to F[4] = 65537.)
a(n) = 3 for n in A000720(A081091) = (4, 5, 6, 8, 12, 13, 19, 21, 25, 32, 33, 44, 98, 106, 116, 136, 174, 191, 310, 313, 319, 565, 568, ...). (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.
Christian Elsholtz, Almost all primes have a multiple of small Hamming weight, arXiv:1602.05974 [math.NT], 2016.
Index entries for sequences related to binary expansion of n

Cf. A035103, A035100, A004676, A090455.
Cf. A027697, A027699
Cf. A180024. - Reinhard Zumkeller, Aug 08 2010
Cf. A072084.
Cf. A159611 (indices of 2s), A000720(A081091) (indices of 3s). - M. F. Hasler, Mar 03 2023

a014499 = a000120 . a000040  -- Reinhard Zumkeller, Feb 10 2013

[&+Intseq(NthPrime(n), 2): n in [1..100] ]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 2], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

A014499(n)=hammingweight(prime(n)) \\ M. F. Hasler, Nov 20 2009, updated Mar 03 2023

from sympy import prime
def A014499(n): return prime(n).bit_count() # Chai Wah Wu, Mar 22 2023

[sum(i.digits(base=2)) for i in primes_first_n(200)] # Tom Edgar, Mar 25 2014

a(n) = A000120(A000040(n)).
a(A049084(A061712(n))) = n. - Reinhard Zumkeller, Feb 10 2013
a(n) = [x^prime(n)] (1/(1 - x))*Sum_{k>=0} x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Mar 27 2018

A027697 Odious primes: primes with odd number of 1's in binary expansion.

Original entry on oeis.org

2, 7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 73, 79, 97, 103, 107, 109, 127, 131, 137, 151, 157, 167, 173, 179, 181, 191, 193, 199, 211, 223, 227, 229, 233, 239, 241, 251, 271, 283, 307, 313, 331, 367, 379, 397, 409, 419, 421, 431, 433, 439, 443, 457, 463, 487, 491, 499, 521, 541, 557, 563

Offset: 1

N. J. A. Sloane

Conjecture: a(n) < A027699(n) except for n = 2; verified up to n=5*10^7. Moreover, I conjecture that A027699(n) - a(n) tends to infinity. - Vladimir Shevelev

T. D. Noe, Table of n, a(n) for n=1..10000
E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, (French) [Sums of digits and almost primes] Math. Ann. 305 (1996), no. 3, 571--599. MR1397437 (97k:11029)
Ben Green, Three topics in additive prime number theory, arXiv:0710.0823 [math.NT], Oct 03, 2007, pp. 12-27.
Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12.

Cf. A027699, A066148, A066149, A081092.

Cf. A000069 (odious numbers), A092246 (odd odious numbers)

a:=proc(n) local nn: nn:= convert(ithprime(n), base, 2): if `mod`(sum(nn[j], j =1..nops(nn)), 2)=1 then ithprime(n) else end if end proc: seq(a(n),n=1..103); # Emeric Deutsch, Oct 24 2007

Clear[BinSumOddQ];BinSumOddQ[a_]:=Module[{i,s=0},s=0;For[i=1,i<=Length[IntegerDigits[a,2]],s+=Extract[IntegerDigits[a,2],i];i++ ];OddQ[s]]; lst={};Do[p=Prime[n];If[BinSumOddQ[p],AppendTo[lst,p]],{n,4!}];lst (* Vladimir Joseph Stephan Orlovsky, Apr 06 2009 *)
Select[Prime@ Range@ 120, OddQ@ First@ DigitCount[#, 2] &] (* Michael De Vlieger, Feb 08 2016 *)

f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==1,s++));return(s%2)};
forprime(p=2, 563, if(f(p), print1(p,", "))) \\ Washington Bomfim, Jan 14 2011

s=[]; forprime(p=2, 1000, if(norml2(binary(p))%2==1, s=concat(s, p))); s \\ Colin Barker, Feb 18 2014

from sympy import primerange
print([n for n in primerange(1, 1001) if bin(n)[2:].count("1")%2]) # Indranil Ghosh, May 03 2017

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A091209 Primes whose binary representation encodes a polynomial reducible over GF(2).

Original entry on oeis.org

5, 17, 23, 29, 43, 53, 71, 79, 83, 89, 101, 107, 113, 127, 139, 149, 151, 163, 173, 179, 181, 197, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 281, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367, 373, 383, 389, 401, 409, 421, 431, 439, 443, 449, 457, 461, 467, 479, 491, 503, 509, 521, 523

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

"Encoded in binary representation" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where each coefficient a(k) = 0 or 1).

Except for 3, all primes with even Hamming weight (A027699) are terms, see A238186 for the subsequence of primes with odd Hamming weight. [Joerg Arndt and Antti Karttunen, Feb 19 2014]

Antti Karttunen, Table of n, a(n) for n = 1..71800
A. Karttunen, Scheme-program for computing this sequence.
Index entries for sequences related to binary encoded polynomials over GF(2)

Intersection of A000040 and A091242.
Disjoint union of A238186 and (A027699 \ {3}).
Left inverse: A235043.
Cf. A091206 (Primes whose binary expansion encodes a polynomial irreducible over GF(2)), A091212 (Composite, and reducible over GF(2)), A091214 (Composite, but irreducible over GF(2)).
Cf. also A235041-A235042, A234742.

Primes:= select(isprime,[2,seq(2*i+1,i=1..1000)]):
filter:= proc(n) local L,x;
    L:= convert(n,base,2);
    Irreduc(add(L[i]*x^(i-1),i=1..nops(L))) mod 2;
end proc:
remove(filter,Primes); # Robert Israel, May 17 2015

Select[Prime[Range[2, 100]], !IrreduciblePolynomialQ[bb = IntegerDigits[#, 2]; Sum[bb[[k]] x^(k-1), {k, 1, Length[bb]}], Modulus -> 2]&] (* Jean-François Alcover, Feb 28 2016 *)

forprime(p=2, 10^3, if( ! polisirreducible( Mod(1,2)*Pol(binary(p)) ), print1(p,", ") ) ); \\ Joerg Arndt, Feb 19 2014

a(n) = A000040(A091210(n)) = A091242(A091211(n)).
Other identities. For all n >= 1:
A235043(a(n)) = n. [A235043 works as a left inverse of this sequence.]

cp's OEIS Frontend

A095006 Number of evil primes (A027699) in range ]2^n,2^(n+1)].

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A177748 First primes of record chains of consecutive primes such that all of them are evil (A027699).

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A230353 Products of 3 evil primes (A027699) p,q,r, such that numbers p*q, p*r, q*r, and p*q*r are odious (A000069).

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A231254 Odious primes p (A027697) such that p^2 and p^3 are evil (A027699).

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A360648 Fully multiplicative with a(A027697(k)) = A027699(k) and a(A027699(k)) = A027697(k) for any k > 0.

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A131122 Even numbers that are not the sum of an evil prime (A027699) and an odious prime (A027697).

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A001969 Evil numbers: nonnegative integers with an even number of 1's in their binary expansion.

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A230353 Products of 3 evil primes (A027699) p,q,r, such that numbers pq, pr, qr, and pq*r are odious (A000069).