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A137985 Complementing any single bit in the binary representation of these primes produces a composite number. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

REFERENCES

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).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Warren D. Smith et al., Primes such that every bit matters?, Yahoo group "primenumbers", April 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. [Cached copy]

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

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.

MAPLE

P:=proc(n) local a, b, c, d, j, k, ok; a:=ithprime(n);

b:=convert(a, base, 2); ok:=1; for k from 1 to nops(b) do

c:=b; c[k]:=(c[k]+1) mod 2; d:=0;

for j from 1 to nops(c) do d:=2*d+c[-j]; od;

if isprime(d) then ok:=0; break; fi; od; if ok=1 then a; fi; end:

seq(P(i), i=1..420); # Paolo P. Lava, Dec 24 2018

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A050249 (analogous base 10 sequence), A186995 (weak primes in base n).

A065092 is a very similar sequence.

Sequence in context: A094933 A156702 A180536 * A065092 A141916 A023689

Adjacent sequences:  A137982 A137983 A137984 * A137986 A137987 A137988

KEYWORD

nonn,base

AUTHOR

T. D. Noe, Feb 26 2008

EXTENSIONS

Definition clarified by Chai Wah Wu, Jan 03 2019

STATUS

approved

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Last modified February 28 06:55 EST 2020. Contains 332321 sequences. (Running on oeis4.)