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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 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=(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 * A297254 A296695 A228948 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 July 2 14:41 EDT 2020. Contains 335401 sequences. (Running on oeis4.)