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%I #71 Feb 21 2024 01:46:13
%S 127,173,191,223,233,239,251,257,277,337,349,373,431,443,491,509,557,
%T 653,683,701,733,761,787,853,877,1019,1193,1201,1259,1381,1451,1453,
%U 1553,1597,1709,1753,1759,1777,1973,2027,2063,2333,2371,2447,2633,2879,2917
%N Complementing any single bit in the binary representation of these primes produces a composite number.
%C 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 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 Digitally delicate primes in base 2. - _Marc Morgenegg_, Apr 21 2021
%D Cohen, Fred; Selfridge, J. L., Not every number is the sum or difference of two prime powers. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 79-81. MR0376583 (51 #12758).
%H T. D. Noe, <a href="/A137985/b137985.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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 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.
%e 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.
%t 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 isWPbase2[z_] := NestWhile[#*2 &, 2, (# < z && ! PrimeQ@BitXor[z, #] &)] > z; Select[Prime /@ Range[3, PrimePi[10^6]], isWPbase2@# &] (* _Terentyev Oleg_, Jul 17 2011 *)
%o (PARI)f(p)={pow2=1;v=binary(p);L=#v;
%o 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 (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 (Python)
%o from sympy import isprime, primerange
%o def ok(p): # p assumed prime
%o return not any(isprime((1<<k)^p) for k in range(p.bit_length()))
%o def aupto(limit):
%o alst = []
%o for p in primerange(2, limit+1):
%o if ok(p): alst.append(p)
%o return alst
%o print(aupto(2917)) # _Michael S. Branicky_, Feb 16 2021
%Y Cf. A050249 (analogous base 10 sequence), A186995 (weak primes in base n).
%Y A065092 is a very similar sequence.
%K nonn,base
%O 1,1
%A _T. D. Noe_, Feb 26 2008
%E Definition clarified by _Chai Wah Wu_, Jan 03 2019
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