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A384021 Powers of 2 along with numbers one power of 2 less than binary repunits, but the power of two subtracted does not flip the leading bit.

%I A384021 #43 Jun 13 2025 08:20:28
%S A384021 1,2,4,5,6,8,11,13,14,16,23,27,29,30,32,47,55,59,61,62,64,95,111,119,
%T A384021 123,125,126,128,191,223,239,247,251,253,254,256,383,447,479,495,503,
%U A384021 507,509,510,512,767,895,959,991,1007,1015,1019,1021,1022,1024,1535,1791,1919
%N A384021 Powers of 2 along with numbers one power of 2 less than binary repunits, but the power of two subtracted does not flip the leading bit.
%C A384021 Also numbers where in the binary expansion the bit 0 or 1 occurs exactly once.
%C A384021 Union of A000079 and A164874. - _Chai Wah Wu_, May 21 2025
%H A384021 David A. Corneth, <a href="/A384021/b384021.txt">Table of n, a(n) for n = 1..10011</a>
%e A384021 4 = 2^2 is term as it is a power of 2.
%e A384021 5 is a term as 5 = (2^3 - 1) - 2^1; a power of two less than a binary repunit and subtracting 2 from 7 does not flip the most significant bit of 7.
%t A384021 A384021list[k_] := Flatten[{1, Table[{2^i - 1 - BitShiftRight[2^i, Range[2, i]], 2^i}, {i, 2 - Boole[k == 1], k}]}]; (* returns terms up to 2^k *)
%t A384021 A384021list[11] (* _Paolo Xausa_, Jun 12 2025 *)
%o A384021 (PARI) upto(n) = {
%o A384021     my(res = List());
%o A384021     for(i = 0, logint(n, 2)+1,
%o A384021         pow2 = 1<<i;
%o A384021         listput(res, pow2);
%o A384021             for(j = 0, i-2,
%o A384021                 listput(res, pow2 - 1<<j - 1);
%o A384021             );
%o A384021         );	
%o A384021         Set(res)}
%o A384021 (Python)
%o A384021 from itertools import count, islice
%o A384021 def agen(): # generator of terms
%o A384021     yield from (1, 2)
%o A384021     for d in count(3):
%o A384021         m, b1 = 1<<(d-1), (1<<d) - 1
%o A384021         yield m
%o A384021         yield from (b1-(m>>i) for i in range(1, d))
%o A384021 print(list(islice(agen(), 58))) # _Michael S. Branicky_, May 18 2025
%o A384021 (Python)
%o A384021 def A384021(n):
%o A384021     def bisection(f,kmin=0,kmax=1):
%o A384021         while f(kmax) > kmax: kmax <<= 1
%o A384021         kmin = kmax >> 1		
%o A384021         while kmax-kmin > 1:
%o A384021             kmid = kmax+kmin>>1
%o A384021             if f(kmid) <= kmid:
%o A384021                 kmax = kmid
%o A384021             else:
%o A384021                 kmin = kmid
%o A384021         return kmax
%o A384021     def f(x):
%o A384021         if x<=1: return n
%o A384021         l, s = x.bit_length(), bin(x)[2:]
%o A384021         if (m:=s.count('0'))>0: return n+x-s.index('0')+(m>1)-(l*(l-1)>>1)
%o A384021         return n+x+1-(l*(l+1)>>1)
%o A384021     return bisection(f,n,n) # _Chai Wah Wu_, May 21 2025
%Y A384021 Cf. A383666 (complement), A000079, A030130, A164874.
%K A384021 nonn,base,easy
%O A384021 1,2
%A A384021 _David A. Corneth_, May 17 2025