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A091936 Smallest prime between 2^n and 2^(n+1), having a minimal number of 1's in binary representation. 7
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A027426 A133928 A126204 * A153145 A174003 A144572
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 14 2004
EXTENSIONS
More terms from Robert G. Wilson v, Feb 18 2004
STATUS
approved

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Last modified June 12 16:34 EDT 2024. Contains 373334 sequences. (Running on oeis4.)