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A091938 Smallest prime between 2^n and 2^(n+1), having a maximal number of 1's in binary representation. 7

%I #13 Apr 09 2020 00:56:02

%S 3,7,11,31,47,127,191,383,991,2039,3583,8191,15359,20479,63487,131071,

%T 245759,524287,786431,1966079,4128767,7323647,14680063,33546239,

%U 67108351,100646911,260046847,536739839,1073479679,2147483647

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

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

%H Chai Wah Wu, <a href="/A091938/b091938.txt">Table of n, a(n) for n = 1..1000</a>

%t 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 *)

%t 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_ *)

%o (Python)

%o from sympy import isprime

%o from sympy.utilities.iterables import multiset_permutations

%o def A091938(n):

%o for i in range(n,-1,-1):

%o q = 2**n

%o for d in multiset_permutations('0'*(n-i)+'1'*i):

%o p = q+int(''.join(d),2)

%o if isprime(p):

%o return p # _Chai Wah Wu_, Apr 08 2020

%Y Cf. A091936, A000668.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Feb 14 2004

%E More terms from _Robert G. Wilson v_, Feb 20 2004

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)