A027697 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

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 *)

PROG

(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

CROSSREFS

Cf. A027699, A066148, A066149, A081092.

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

Sequence in context: A161681 A020583 A140557 * A364470 A235475 A146315