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A137985
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Complementing any single bit in the binary representation of these primes produces a composite number.
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10
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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
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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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).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
isWPbase2[z_] := NestWhile[#*2 &, 2, (# < z && ! PrimeQ@BitXor[z, #] &)] > z; Select[Prime /@ Range[3, PrimePi[10^6]], isWPbase2@# &] (* Terentyev Oleg, Jul 17 2011 *)
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PROG
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(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<<k)^p) for k in range(p.bit_length()))
def aupto(limit):
alst = []
for p in primerange(2, limit+1):
if ok(p): alst.append(p)
return alst
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CROSSREFS
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Cf. A050249 (analogous base 10 sequence), A186995 (weak primes in base n).
A065092 is a very similar sequence.
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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