A225781 Numbers k such that both k and (k+1)/2 are primes and evil.

5, 277, 673, 1093, 1237, 1381, 1621, 1873, 2473, 2593, 2797, 2857, 4177, 4357, 4441, 4561, 4933, 5077, 5233, 5413, 5437, 5581, 5701, 6037, 6133, 6997, 7477, 7537, 8053, 8353, 8713, 8893, 9133, 9901, 10861, 10957, 11113, 11161, 11497, 12073, 12457, 12757

Offset: 1

Author

Brad Clardy, May 15 2013

Keywords

• nonn
• base

Comments

It seems to be the case that all primes k where (k+1)/2 is also prime share the property that they are also both either evil or odious, the sole exception being 3, which is evil but has 2 as an odious companion.

The last comment is true; for k and (k+1)/2 to be prime, k must be the number 3 or have the form 4*m + 1. The latter means its binary expansion ends in 01. Adding 1 to such a number and dividing by 2 leaves the bit count the same. Hence, both of these numbers have the same parity; they are both evil or both odious. - Jon Perry, May 25 2013

Links

• Brad Clardy, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A005383 (both k and (k+1)/2 are primes), A001969 (evil numbers).

Programs

Magma

//the function Bweight determines the binary weight of a number
Bweight := function(m)
Bweight:=0;
adigs := Intseq(m,2);
for n:= 1 to Ilog2(m)+1 do
  Bweight:=Bweight+adigs[n];
end for;
return Bweight;
end function;
for i:=1 to 1000000 do
pair:=(i+1)div 2;
  if (IsPrime(i) and IsPrime(pair) and (Bweight(i) mod 2 eq 0) and     (Bweight(pair) mod 2 eq 0)) then i;
  end if;
end for;

Mathematica

evilQ[n_] := EvenQ[DigitCount[n, 2, 1]]; Select[Prime[Range[1600]], PrimeQ[(#+1)/2] && And @@ evilQ /@ {#, (#+1)/2} &] (* Amiram Eldar, Aug 06 2023 *)