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

1,1

Comments

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

Links

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

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 *)

Prog

(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

Crossrefs

Cf. A091936, A000668.

Sequence in context: A274134 A131588 A361680 * A333424 A186893 A051919

Adjacent sequences: A091935 A091936 A091937 * A091939 A091940 A091941

Keyword

nonn

Author

Reinhard Zumkeller, Feb 14 2004

Extensions

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

Status

approved