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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

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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