A066148 -id:A066148 - cp's OEIS frontend

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

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)

A027699 Evil primes: primes with even number of 1's in their binary expansion.

Original entry on oeis.org

3, 5, 17, 23, 29, 43, 53, 71, 83, 89, 101, 113, 139, 149, 163, 197, 257, 263, 269, 277, 281, 293, 311, 317, 337, 347, 349, 353, 359, 373, 383, 389, 401, 449, 461, 467, 479, 503, 509, 523, 547, 571, 593, 599, 619, 643, 673, 683, 691, 739, 751, 773, 797, 811

Offset: 1

N. J. A. Sloane

Comment from Vladimir Shevelev, Jun 01 2007: Conjecture: If pi_1(m) is the number of a(n) not exceeding m and pi_2(m) is the number of A027697(n) not exceeding m then pi_1(m) <= smaller than pi_2(m) for all natural m except m=5 and m=6. I verified this conjecture up to 10^9. Moreover I conjecture that pi_2(m)-pi_1(m) tends to infinity with records at the primes m=2, 13, 41, 61, 67, 79, 109, 131, 137, ...

T. D. Noe, Table of n, a(n) for n = 1..10000
E. Fouvry, 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).
V. Shevelev, A conjecture on primes and a step towards justification, arXiv:0706.0786 [math.NT], 2007.

Cf. A027697, A066148, A066149.
Cf. A001969 (evil numbers), A129771 (evil odd numbers)
Cf. A130911 (prime race between evil primes and odious primes).

Select[Prime[Range[200]], EvenQ[Count[IntegerDigits[ #,2],1]]&] (* T. D. Noe, Jun 12 2007 *)

forprime(p=1,999,norml2(binary(p))%2 || print1(p","))

isA027699(p)=isprime(p) && !bittest(norml2(binary(p)),0) \\ M. F. Hasler, Dec 12 2010

from sympy import isprime
def ok(n): return bin(n).count("1")%2 == 0 and isprime(n)
print([k for k in range(812) if ok(k)]) # Michael S. Branicky, Jun 27 2022

More terms from Erich Friedman

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

A156549 Race between primes having an odd/even number of zeros in their binary representation.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 6, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 3, 4, 5, 4, 3, 4, 5, 4, 5, 6, 7, 8, 9, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 21, 22, 21, 22, 21, 22, 21, 22, 21, 22, 23, 24, 25, 26, 25, 26, 25, 26, 27, 26, 27, 26, 25, 24, 23, 22

Offset: 1

T. D. Noe, Feb 09 2009

See A066148 and A066149 for primes with an even/odd number of zeros in their binary representation. Sequence A130911 shows the race between primes having an odd/even number of ones in their binary representation. In this sequence (and A130911), it appears that the primes with an odd number of zeros (or ones) dominate the primes with an even number of zeros (or ones). In general, it appears that the sequences grow for primes having an odd number of bits and "rest" for primes having an even number of bits.

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

Cf. A066148, A066149, A130911.

cnt=0; Table[p=Prime[n]; If[OddQ[Count[IntegerDigits[p,2],0]], cnt++, cnt-- ]; cnt, {n,100}]
Accumulate[Table[If[OddQ[DigitCount[p,2,0]],1,-1],{p,Prime[Range[90]]}]] (* Harvey P. Dale, Apr 02 2025 *)

f(p)={v=binary(p);s=0;for(k=1,#v,if(v[k]==0, s++));return(s%2)}; nO=0;nE=0; forprime(p=2,435,if(f(p), nO++, nE++); an = nO-nE; print1(an,", ")) \\ Washington Bomfim, Jan 14 2011

a(n) = (number of primes having an odd number of zeros <= prime(n)) - (number of primes having an even number of zeros <= prime(n))

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

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A027699 Evil primes: primes with even number of 1's in their binary expansion.

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

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A156549 Race between primes having an odd/even number of zeros in their binary representation.

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