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

Offset

1,1

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

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]

C. Rivera (Ed.), Problem 25: William Paulsen's Prime Numbers Maze, on primepuzzles.net. [Before Oct. 1999]

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.

Example

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.

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} ]

Prog

(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<<k)^p) for k in range(p.bit_length()+1))
def aupto(limit):
    alst = []
    for p in primerange(2, limit+1):
        if ok(p): alst.append(p)
    return alst
print(aupto(2917)) # Michael S. Branicky, Jul 26 2022

Crossrefs

Sequence in context: A180536 A342801 A137985 * A141916 A023689 A095284

Adjacent sequences: A065089 A065090 A065091 * A065093 A065094 A065095

Keyword

base,nonn

Author

Robert G. Wilson v, Nov 10 2001

Extensions

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

Status

approved