A091936 Smallest prime between 2^n and 2^(n+1), having a minimal number of 1's in binary representation.

2, 5, 11, 17, 37, 67, 131, 257, 521, 1033, 2053, 4099, 8209, 16417, 32771, 65537, 133121, 262147, 524353, 1048609, 2097169, 4194433, 8388617, 16777729, 33554467, 67239937, 134250497, 268435459, 536903681, 1073741827, 2147483713

Offset

1,1

Comments

A091935(n) = A000120(a(n)).

So far only a(25) and a(32) possess 4 1's in their binary representation.

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 = Infinity; While[ p < 2^n, b = Count[ IntegerDigits[ p, 2], 1]; If[ c > b, c = b; q = p]; p = NextPrim[ p]; If[ c < 4, p = NextPrim[ 2^n]; Continue[ ]]]; Print[ q], {n, 2, 32}] (* Robert G. Wilson v, Feb 18 2004 *)
b[ n_ ] := Min[ Select[ FromDigits[ #, 2 ] & /@ (Join[ {1}, #, {1} ] & /@ Permutations[ Join[ {1}, Table[ 0, {n - 2} ] ] ]), PrimeQ[ # ] & ] ]; c[ n_ ] := Min[ Select[ FromDigits[ #, 2 ] & /@ (Join[ {1}, #, {1} ] & /@ Permutations[ Join[ {1, 1}, Table[ 0, {n - 3} ] ] ]), PrimeQ[ # ] & ] ]; f[ n_ ] := If[ PrimeQ[ 2^n + 1 ], 2^n + 1, If[ PrimeQ[ b[ n ] ], b[ n ], c[ n ] ] ]; Table[ f[ n ], {n, 2, 32} ] (* Robert G. Wilson v *)

Prog

(Python)
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def A091936(n):
    for i in range(n+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. A091938, A019434.

Sequence in context: A027426 A133928 A126204 * A153145 A174003 A144572

Adjacent sequences: A091933 A091934 A091935 * A091937 A091938 A091939

Keyword

nonn

Author

Reinhard Zumkeller, Feb 14 2004

Extensions

More terms from Robert G. Wilson v, Feb 18 2004

Status

approved