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

2,1

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A006978 A060424 A119962 * A220746 A110436 A126879

Adjacent sequences: A333874 A333875 A333876 * A333878 A333879 A333880

Keyword

nonn

Author

Hugo Pfoertner, Apr 08 2020

Status

approved