A091938 Smallest prime between 2^n and 2^(n+1), having a maximal number of 1's in binary representation.

3, 7, 11, 31, 47, 127, 191, 383, 991, 2039, 3583, 8191, 15359, 20479, 63487, 131071, 245759, 524287, 786431, 1966079, 4128767, 7323647, 14680063, 33546239, 67108351, 100646911, 260046847, 536739839, 1073479679, 2147483647

Offset: 1

Author

Reinhard Zumkeller, Feb 14 2004

Keywords

• nonn

Comments

A091937(n) = A000120(a(n)).

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A091936, A000668.

Programs

Mathematica

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 2; Do[c = 0; While[p < 2^n, b = Count[ IntegerDigits[p, 2], 1]; If[c < b, c = b; q = p]; p = NextPrim[p]]; Print[q], {n, 1, 30}] (* Robert G. Wilson v, Feb 21 2004 *)
b[n_] := Min[ Select[ FromDigits[ #, 2] & /@ (Join[{1}, #, {1}] & /@ Permutations[ Join[{0}, Table[1, {n - 2}]]]), PrimeQ[ # ] &]]; c[n_] := Min[ Select[ FromDigits[ #, 2] & /@ (Join[{1}, #, {1}] & /@ Permutations[ Join[{0, 0}, Table[1, {n - 3}]]]), PrimeQ[ # ] &]]; f[n_] := If[ PrimeQ[2^(n + 1) - 1], 2^(n + 1) - 1, If[ PrimeQ[ b[n]], b[n], c[n]]]; Table[ f[n], {n, 30}] (* Robert G. Wilson v *)

Python

from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A091938(n):
    for i in range(n,-1,-1):
        q = 2**n
        for d in multiset_permutations('0'*(n-i)+'1'*i):
            p = q+int(''.join(d),2)
            if isprime(p):
                return p # Chai Wah Wu, Apr 08 2020

Extensions

More terms from Robert G. Wilson v, Feb 20 2004