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A267413 Dropping any binary digit gives a prime number. 1
6, 7, 11, 15, 35, 39, 63, 135, 255, 999, 2175, 8223, 16383, 57735, 131075, 131079, 262143, 524295, 1048575, 536870919, 1073735679, 2147483655, 4294967295, 17179770879 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is the binary analog of A034895. The sequence contains mostly numbers with very few binary digit runs (BDR, A005811). Those with one BDR are of the type 2^k-1, such that 2^(k-1)-1 is a Mersenne prime (A000668). Vice versa, if M is any Mersenne prime, then 2*M+1 is a member. Number 6 is the only member with an even number of BDRs. There are many members with 3 BDRs. The first member with 5 BDRs is 57735. Next members with at least 5 BDRs (if they exist at all) are larger than 10^10. So far, I could test that a(24)>10^10.

From Robert Israel, Jan 14 2016: (Start)

For n >= 2, a(n) == 3 (mod 4).

2^k+3 is in the sequence if 2^(k-1)+1 and 2^(k-1)+3 are primes, i.e. 2^(k-1)+1 is in the intersection of A019434 and A001359.  The only known members of the sequence in this class are 7, 11, 35, 131075.

2^k+7 is in the sequence if 2^(k-1)+3 and 2^(k-1)+7 are primes: thus 2^(k-1)+3 is in A057733 and 2^(k-1)+7 is in A104066.  Members of the sequence in this class include 15, 39, 135, 131079, 524295, 536870919, 2147483655 (but no more for k <= 2000).

(End)

a(25) > 2^38. - Giovanni Resta, Apr 10 2016

LINKS

Table of n, a(n) for n=1..24.

EXAMPLE

Decimal and binary forms of the first few members:

1  6           110

2  7           111

3  11          1011

4  15          1111

5  35          100011

6  39          100111

7  63          111111

8  135         10000111

9  255         11111111

10 999         1111100111

11 2175        100001111111

12 8223        10000000011111

13 16383       11111111111111

14 57735       1110000110000111 (binary palindrome with 5 digit runs)

15 131075      100000000000000011

16 131079      100000000000000111

17 262143      111111111111111111

18 524295      10000000000000000111

19 1048575     11111111111111111111

20 536870919   100000000000000000000000000111

21 1073735679  111111111111111110011111111111

22 2147483655  10000000000000000000000000000111

23 4294967295  11111111111111111111111111111111

24 17179770879 1111111111111111100111111111111111

MAPLE

filter:= proc(n) local B, k, y;

   if not isprime(floor(n/2)) then return false fi;

   B:= convert(n, base, 2);

   for k from 2 to nops(B) do

     if B[k] <> B[k-1] then

       y:= n mod 2^(k-1);

       if not isprime((y+n-B[k]*2^(k-1))/2) then return false fi

     fi

   od;

   true

end proc:

select(filter, [6, seq(i, i=7..10^6, 4)]); # Robert Israel, Jan 14 2016

MATHEMATICA

Select[Range[2^20], AllTrue[Function[w, Map[FromDigits[#, 2] &@ Drop[w, {#}] &, Range@ Length@ w]]@ IntegerDigits[#, 2], PrimeQ] &] (* Michael De Vlieger, Jan 16 2016, Version 10 *)

PROG

(PARI) DroppingAnyDigitGivesAPrime(N, b) = {

\\ Property-testing function; returns 1 if true for N, 0 otherwise

\\ Works with any base b. Here used with b=2.

  my(k=b, m); if(N<b, return(0));

  while(N>=(k\b), m=(N\k)*(k\b)+(N%(k\b));

    if ((m<2)||(!isprime(m)), return(0)); k*=b);

  return(1);

}

CROSSREFS

Cf. A000668, A001359, A005811, A019434, A034895 (base 10), A051362, A057733, A104066.

Sequence in context: A224856 A182156 A166496 * A228948 A035110 A011990

Adjacent sequences:  A267410 A267411 A267412 * A267414 A267415 A267416

KEYWORD

nonn,base,more,hard

AUTHOR

Stanislav Sykora, Jan 14 2016

EXTENSIONS

a(24) from Giovanni Resta, Apr 10 2016

STATUS

approved

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Last modified May 22 16:19 EDT 2017. Contains 286882 sequences.