A137985 Complementing any single bit in the binary representation of these primes does not produce a prime number.

127, 173, 191, 223, 233, 239, 251, 257, 277, 337, 349, 373, 431, 443, 491, 509, 557, 653, 683, 701, 733, 761, 787, 853, 877, 1019, 1193, 1201, 1259, 1381, 1451, 1453, 1553, 1597, 1709, 1753, 1759, 1777, 1973, 2027, 2063, 2333, 2371, 2447, 2633, 2879, 2917, 2999

Offset: 1

Author

T. D. Noe, Feb 26 2008

Keywords

• nonn
• base

Comments

If 2^m is the highest power of 2 in the binary representation of the prime p, there is no requirement that p+2^(m+1) be composite. Sequence A065092 imposes this extra requirement. The prime 223 is the first number in this sequence that is not in A065092.

Mentioned Feb 25 2008 by Terence Tao in his blog http://terrytao.wordpress.com. Tao proves that there are an infinite number of these primes in every fixed base.

Digitally delicate primes in base 2. - Marc Morgenegg, Apr 21 2021

Examples

			The numbers produced by complementing each of the 8 bits of 223 are 95, 159, 255, 207, 215, 219, 221 and 222, which are all composite.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comp. 29 (1975), 79-81.
• Warren D. Smith et al., Primes such that every bit matters?, Yahoo group "primenumbers", April 2013.
• Warren D. Smith and others, Primes such that every bit matters?, digest of 14 messages in primenumbers Yahoo group, Apr 3 - Apr 9, 2013. [Cached copy]
• Terence Tao, A remark on primality testing and decimal expansions, arXiv:0802.3361 [math.NT], 2008-2010; Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
• Eric Weisstein, Weakly Prime. From MathWorld - A Wolfram Web Resource.
• Wikipedia, Delicate prime

Crossrefs

Cf. A050249 (analogous base 10 sequence), A186995 (weak primes in base n).

A065092 is a very similar sequence.

Programs

Maple

q:= p-> isprime(p) and not ormap(i->isprime(Bits[Xor](p, 2^i)), [$0..ilog2(p)]):
select(q, [$2..5000])[];  # Alois P. Heinz, Jul 28 2025

Mathematica

t={}; k=1; While[Length[t]<100, k++; p=Prime[k]; d=IntegerDigits[p,2]; n=Length[d]; i=0; While[iT. D. Noe *)
isWPbase2[z_] := NestWhile[#*2 &, 2, (# < z && ! PrimeQ@BitXor[z, #] &)] > z; Select[Prime /@ Range[3, PrimePi[10^6]], isWPbase2@# &] (* Terentyev Oleg, Jul 17 2011 *)
Select[Prime[Range[500]], NoneTrue[BitXor[#, 2^Range[0, BitLength[#] - 1]], PrimeQ] &] (* Paolo Xausa, Apr 23 2025 *)

PARI

f(p)={pow2=1;v=binary(p);L=#v;
forstep(k=L,1,-1,if(v[k],p-=pow2;if(isprime(p),return(0),p+=pow2),p+=pow2;if(isprime(p),return(0),p-=pow2)); pow2*=2);return(1)}; forprime(p=2,2879,if(f(p), print1(p,", "))) \\ Washington Bomfim, Jan 18 2011

PARI

is_A137985(n)=!for(k=1,n,isprime(bitxor(n,k)) && return;k+=k-1) && isprime(n) \\ Note: A bug in early versions of PARI 2.6 (execute "for(i=0,1,i>3 && error(buggy);i=9)" to check) makes that this is is_A065092 rather than is_A137985 as expected. For these versions, replace the upper limit n with n\2. \\ M. F. Hasler, Apr 05 2013

Python

from sympy import isprime, primerange
def ok(p): # p assumed prime
  return not any(isprime((1<Michael S. Branicky, Feb 16 2021

Extensions

Definition clarified by Chai Wah Wu, Jan 03 2019

Name edited by Paolo Xausa, Apr 24 2025