cp's OEIS Frontend

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

%I A137985 #99 Jul 28 2025 14:30:48
%S A137985 127,173,191,223,233,239,251,257,277,337,349,373,431,443,491,509,557,
%T A137985 653,683,701,733,761,787,853,877,1019,1193,1201,1259,1381,1451,1453,
%U A137985 1553,1597,1709,1753,1759,1777,1973,2027,2063,2333,2371,2447,2633,2879,2917,2999
%N A137985 Complementing any single bit in the binary representation of these primes does not produce a prime number.
%C A137985 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.
%C A137985 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.
%C A137985 Digitally delicate primes in base 2. - _Marc Morgenegg_, Apr 21 2021
%H A137985 T. D. Noe, <a href="/A137985/b137985.txt">Table of n, a(n) for n = 1..10000</a>
%H A137985 Fred Cohen and J. L. Selfridge, <a href="https://doi.org/10.1090/S0025-5718-1975-0376583-0">Not every number is the sum or difference of two prime powers</a>, Math. Comp. 29 (1975), 79-81.
%H A137985 Warren D. Smith et al., <a href="http://groups.yahoo.com/group/primenumbers/message/24989">Primes such that every bit matters?</a>, Yahoo group "primenumbers", April 2013.
%H A137985 Warren D. Smith and others, <a href="/A137985/a137985.txt">Primes such that every bit matters?</a>, digest of 14 messages in primenumbers Yahoo group, Apr 3 - Apr 9, 2013. [Cached copy]
%H A137985 Terence Tao, <a href="http://arxiv.org/abs/0802.3361">A remark on primality testing and decimal expansions</a>, arXiv:0802.3361 [math.NT], 2008-2010; Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
%H A137985 Eric Weisstein, <a href="https://mathworld.wolfram.com/WeaklyPrime.html">Weakly Prime.</a> From MathWorld - A Wolfram Web Resource.
%H A137985 Wikipedia, <a href="https://en.wikipedia.org/wiki/Delicate_prime">Delicate prime</a>
%e A137985 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.
%p A137985 q:= p-> isprime(p) and not ormap(i->isprime(Bits[Xor](p, 2^i)), [$0..ilog2(p)]):
%p A137985 select(q, [$2..5000])[];  # _Alois P. Heinz_, Jul 28 2025
%t A137985 t={}; k=1; While[Length[t]<100, k++; p=Prime[k]; d=IntegerDigits[p,2]; n=Length[d]; i=0; While[i<n && (d[[n-i]]==1 && !PrimeQ[p-2^i]) || (d[[n-i]]==0 && !PrimeQ[p+2^i]), i++ ]; If[i==n, AppendTo[t,p]]]; t  (* _T. D. Noe_ *)
%t A137985 isWPbase2[z_] := NestWhile[#*2 &, 2, (# < z && ! PrimeQ@BitXor[z, #] &)] > z; Select[Prime /@ Range[3, PrimePi[10^6]], isWPbase2@# &] (* _Terentyev Oleg_, Jul 17 2011 *)
%t A137985 Select[Prime[Range[500]], NoneTrue[BitXor[#, 2^Range[0, BitLength[#] - 1]], PrimeQ] &] (* _Paolo Xausa_, Apr 23 2025 *)
%o A137985 (PARI)f(p)={pow2=1;v=binary(p);L=#v;
%o A137985 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
%o A137985 (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
%o A137985 (Python)
%o A137985 from sympy import isprime, primerange
%o A137985 def ok(p): # p assumed prime
%o A137985   return not any(isprime((1<<k)^p) for k in range(p.bit_length()))
%o A137985 def aupto(limit):
%o A137985   alst = []
%o A137985   for p in primerange(2, limit+1):
%o A137985     if ok(p): alst.append(p)
%o A137985   return alst
%o A137985 print(aupto(2917)) # _Michael S. Branicky_, Feb 16 2021
%Y A137985 Cf. A050249 (analogous base 10 sequence), A186995 (weak primes in base n).
%Y A137985 A065092 is a very similar sequence.
%K A137985 nonn,base
%O A137985 1,1
%A A137985 _T. D. Noe_, Feb 26 2008
%E A137985 Definition clarified by _Chai Wah Wu_, Jan 03 2019
%E A137985 Name edited by _Paolo Xausa_, Apr 24 2025