A065092 Primes with property that when written in base two complementing any single bit yields a composite number.

127, 173, 191, 233, 239, 251, 277, 337, 349, 373, 431, 443, 491, 557, 653, 683, 701, 733, 761, 1019, 1193, 1201, 1381, 1453, 1553, 1597, 1709, 1753, 1759, 1777, 2027, 2063, 2333, 2371, 2447, 2633, 2879, 2999, 3083, 3181, 3209, 3313, 3593, 3643, 3767, 3779, 3851

Offset: 1

Author

Robert G. Wilson v, Nov 10 2001

Keywords

• base
• nonn

Comments

Also known as singularly dead end primes.

In contrast to the primes listed in A137985 (which contains, e.g., the additional term 223), the terms listed here are required to yield a composite also when prefixed with an ("additional") binary digit 1. - M. F. Hasler, Apr 05 2013

Examples

			127 is in the sequence because 127d becomes 1111111b. "Changing a 1 to a 0 [from right to left] yields rooms 126, 125, 123, 119, 111, 95, or 62, all of which are composite. Furthermore, adding a digit 1 to the left of this number produces, 255 = 11111111b which is also composite. However, this room is not completely isolated from the maze because one can drop in from room 383d = 101111111b." Paulsen.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• William Paulsen, Are some rooms totally isolated? [Copy on web.archive.org, latest version as of Nov 04 2008]
• Carlos Rivera, Problem 25: William Paulsen's Prime Numbers Maze, The Prime Puzzles & Problems Connection.
• Warren D. Smith et al., Primes such that every bit matters?, on "primenumbers" Yahoo group, Apr 04 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.
• 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.

Crossrefs

Cf. A137985.

Programs

Maple

q:= p-> isprime(p) and not ormap(i-> isprime(Bits[Xor](p, 2^i)), [$0..ilog2(p)+1]):
select(q, [$2..5000])[];  # Alois P. Heinz, Jul 28 2025

Mathematica

Do[d = Prepend[ IntegerDigits[ Prime[n], 2], 0]; l = Length[d]; k = 1; While[k < l && !PrimeQ[ FromDigits[ If[d[[k]] == 1, ReplacePart[d, 0, k], ReplacePart[d, 1, k]], 2]], k++ ]; If[k == l, Print[ Prime[n]]], {n, 2, 500} ]

PARI

f(p)=
{
  pow2=2;  v=binary(p); L=#v-1;
  forstep(k=L,1,-1,
    if(v[k]==0, x=p+pow2, x=p-pow2);
    if(isprime(x), return(0));
    pow2*=2
  );
  if(isprime(p+pow2), return(0)); return(1)
};
forprime(p=5,3767, if(f(p), print1(p, ", "))) \\ Washington Bomfim, Jan 16 2011

PARI

/* needs ver. >= 2.6 */ is_A065092(n)={!for(k=1,n,isprime(bitxor(n,k))&return;k+=k-1)&isprime(n)} \\ Note the strange behavior of the for() loop w.r.t. the upper limit. In PARI versions up to 2.4, the increment must take place at the beginning of the loop, viz "k>2 & k+=k-2" BEFORE isprime(), as to cover k=2^ceil(log[2](n)). - 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, Jul 26 2022

Extensions

Links fixed & added by M. F. Hasler, Apr 05 2013