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A380788 Numbers with a prime number of binary digits.

%I A380788 #21 Feb 07 2025 05:23:51
%S A380788 2,3,4,5,6,7,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,64,65,66,
%T A380788 67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,
%U A380788 90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106
%N A380788 Numbers with a prime number of binary digits.
%H A380788 Michael S. Branicky, <a href="/A380788/b380788.txt">Table of n, a(n) for n = 1..10000</a>
%e A380788 4 is a term since its binary representation has 3 bits, a prime.
%e A380788 64 is a term since its binary representation has 7 bits, a prime.
%t A380788 Select[Range[200], PrimeQ[BitLength[#]] &] (* _Paolo Xausa_, Feb 03 2025 *)
%o A380788 (Python)
%o A380788 from sympy import isprime
%o A380788 def ok(n): return isprime(n.bit_length())
%o A380788 print([k for k in range(150) if ok(k)])
%o A380788 (Python) # faster for initial segment of sequence
%o A380788 from itertools import islice
%o A380788 from sympy import isprime, nextprime
%o A380788 def agen(): # generator of terms
%o A380788     d = 2
%o A380788     while True:
%o A380788         yield from (i for i in range(2**(d-1), 2**d))
%o A380788         d = nextprime(d)
%o A380788 print(list(islice(agen(), 65)))
%o A380788 (Python)
%o A380788 from sympy import primerange
%o A380788 def A380788(n):
%o A380788     def bisection(f,kmin=0,kmax=1):
%o A380788         while f(kmax) > kmax: kmax <<= 1
%o A380788         kmin = kmax >> 1
%o A380788         while kmax-kmin > 1:
%o A380788             kmid = kmax+kmin>>1
%o A380788             if f(kmid) <= kmid:
%o A380788                 kmax = kmid
%o A380788             else:
%o A380788                 kmin = kmid
%o A380788         return kmax
%o A380788     def f(x): return n+x-sum(min(x,(1<<i)-1)-(1<<i-1)+1 for i in primerange(2,x.bit_length()+1))
%o A380788     return bisection(f,n,n) # _Chai Wah Wu_, Feb 03 2025
%Y A380788 Cf. A272441, A053738.
%K A380788 nonn,base,easy
%O A380788 1,1
%A A380788 _Michael S. Branicky_, Feb 03 2025