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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• base

Comments

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

Links

• 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.

Crossrefs

Cf. A027699, A066148, A066149, A081092.

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

Programs

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Python

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

Extensions

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