A333877 a(n) is the largest prime 2^(n-1) <= p < 2^n maximizing the Hamming weight of all primes in this interval.

3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, 65519, 131071, 262139, 524287, 1048573, 2097143, 4194301, 8388587, 16777213, 33546239, 67108859, 134217467, 260046847, 536870909, 1073741567, 2147483647, 4294967291, 8589934583, 16911433727

Offset: 2

Author

Hugo Pfoertner, Apr 08 2020

Keywords

• nonn

Comments

This differs from A014234 at n=1 and then first at n=16: a(16) = 65519 != 65521 = A014234(16). - Alois P. Heinz, Apr 25 2020

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..998

Crossrefs

Cf. A014234, A091937, A091938, A333876, A333879.

Programs

Maple

a:= proc(n) option remember; local i, p;
      for i from 0 do p:= max(select(isprime, map(l-> add(l[j]*
        2^(j-1), j=1..n), combinat[permute]([1$(n-i),0$i]))));
        if p>0 then break fi
      od; p
    end:
seq(a(n), n=2..30);  # Alois P. Heinz, Apr 23 2020

Mathematica

a[n_] := a[n] = MaximalBy[{#, DigitCount[#, 2, 1]}& /@ Select[Range[ 2^(n-1), 2^n-1], PrimeQ], Last][[-1, 1]];
Table[Print[n, " ", a[n]]; a[n], {n, 2, 30}] (* Jean-François Alcover, Nov 09 2020 *)

PARI

for(n=2, 30, my(hmax=0,pmax); forprime(p=2^(n-1), 2^n, my(h=hammingweight(p)); if(h>=hmax,pmax=p;hmax=h)); print1(pmax,", "))

Python

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A333877(n):
    for i in range(n-1,-1,-1):
        q = 2**n-1
        for d in multiset_permutations('0'*i+'1'*(n-1-i)):
            p = q-int(''.join(d),2)
            if isprime(p):
                return p # Chai Wah Wu, Apr 08 2020