A145576 a(n) is the smallest prime with both exactly an n number of 0's and exactly an n number of 1's in its binary representation. a(n) = 0 if no such prime exists.

2, 0, 37, 139, 541, 2141, 8287, 33119, 131519, 525247, 2098687, 8391679, 33561599, 134242271, 536895487, 2147548159, 8590061567, 34360196863, 137439412223, 549756861439, 2199026663423, 8796097216447, 35184380411903

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

a(3) = 37 = 100101 (base 2) is the smallest prime with three 0's and three 1's in its binary representation. - R. J. Mathar, Oct 14 2008

Maple

A000120 := proc(n) local d; add(d, d=convert(n, base, 2)) ; end: A080791 := proc(n) local d, dgs; dgs := convert(n, base, 2) ; nops(dgs)-add(d, d=dgs) ; end: A070939 := proc(n) max(1, ilog2(n)+1) ; end: A145576 := proc(n) local p, pbin; p := nextprime(2^(2*n-1)-1); while true do pbin := A070939(p) ; if pbin > 2*n then RETURN(0) ; elif pbin = 2*n then if A000120(p) = n and A080791(p) = n then RETURN(p) ; fi; fi; p := nextprime(p) ; od: end: seq(A145576(n), n=1..30) ; # R. J. Mathar, Oct 14 2008
# Alternative:
F:= proc(n) local c, x;
      c:= [$n+1..2*n-2];
      do
        x:= 2^(2*n-1)+1+add(2^(2*n-1-c[i]), i=1..n-2);
        if isprime(x) then return x fi;
        c:= combinat:-prevcomb(c, 2*n-2)
      od
end proc:
2, 0, seq(F(n), n=3..30); # Robert Israel, Sep 24 2017

Mathematica

Table[SelectFirst[Prime@ Apply[Range, PrimePi@{2^(2 (n - 1)) + 1, 2^(2 n) - 1}], Union@ DigitCount[#, 2] == {n} &] /. k_ /; MissingQ@ k -> 0, {n, 12}] (* Michael De Vlieger, Sep 24 2017 *)

Crossrefs

Cf. A066195, A061712.

Sequence in context: A347900 A337817 A120489 * A178235 A214447 A357541

Adjacent sequences: A145573 A145574 A145575 * A145577 A145578 A145579

Keyword

base,nonn

Author

Leroy Quet, Oct 13 2008

Extensions

Extended by R. J. Mathar and Ray Chandler, Oct 14 2008

Status

approved