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A383666 Numbers in whose binary representation no bit (0 or 1) occurs exactly once.

%I A383666 #34 May 21 2025 12:40:18
%S A383666 3,7,9,10,12,15,17,18,19,20,21,22,24,25,26,28,31,33,34,35,36,37,38,39,
%T A383666 40,41,42,43,44,45,46,48,49,50,51,52,53,54,56,57,58,60,63,65,66,67,68,
%U A383666 69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86
%N A383666 Numbers in whose binary representation no bit (0 or 1) occurs exactly once.
%C A383666 Also numbers that are not a power of 2 and are not (2^k + 1) away from the next larger power of 2 for some k. - _David A. Corneth_, May 17 2025
%H A383666 David A. Corneth, <a href="/A383666/b383666.txt">Table of n, a(n) for n = 1..10000</a>
%e A383666 From _David A. Corneth_, May 17 2025: (Start)
%e A383666 3 = 11_2 is in the sequence as both the digits 0 and the digits 1 do not occur exactly once in the binary expansion. Also 3 is no power of 2 and one less than a power of 2.
%e A383666 6 = 101_2 is not in the sequence as the digit 0 occurs exactly once in the binary expansion. Also it can be written as 2^3 - 2^0 - 1. (End)
%p A383666 filter:= proc(n) local L,n1,n0;
%p A383666   L:= convert(n,base,2);
%p A383666   n1:= convert(L,`+`);
%p A383666   n0:= nops(L)-n1;
%p A383666   n1 >= 2 and n0 <> 1
%p A383666 end proc;
%p A383666 select(filter, [$1..1000]); # _Robert Israel_, May 13 2025
%t A383666 s = Select[Range[200], DigitCount[#, 2, 0] != 1 && DigitCount[#, 2, 1] != 1 &]
%t A383666 Map[First, RealDigits[s, 2]]
%o A383666 (PARI) isok(k) = my(b=binary(k)); (#select(x->(x==1), b) != 1) && (#select(x->(x==0), b) != 1); \\ _Michel Marcus_, May 13 2025
%o A383666 (PARI) is(n) = {
%o A383666     my(v = valuation(n, 2));
%o A383666     if(n >> v == 1, return(0));
%o A383666     if(1<<valuation(n+1,2) == n+1, return(1));
%o A383666     c = 1 << (logint(n, 2) + 1) - n - 1;
%o A383666     if(c >> valuation(c, 2) == 1, return(0));
%o A383666     1
%o A383666 } \\ _David A. Corneth_, May 17 2025
%o A383666 (PARI) upto(n) = {
%o A383666     my(res = [1..n], del = List());
%o A383666     for(i = 0, logint(n, 2)+1,
%o A383666         pow2 = 1<<i;
%o A383666         listput(del, pow2);
%o A383666         for(j = 0, i-2,
%o A383666             listput(del, pow2 - 1<<j - 1);
%o A383666         );
%o A383666     );	
%o A383666     setminus(res, Set(del));
%o A383666 } \\ _David A. Corneth_, May 17 2025
%o A383666 (Python)
%o A383666 def A383666(n):
%o A383666     def f(x):
%o A383666         if x<=1: return n+x
%o A383666         l, s = x.bit_length(), bin(x)[2:]
%o A383666         if (m:=s.count('0'))>0: return n+s.index('0')-(m>1)+(l*(l-1)>>1)
%o A383666         return n-1+(l*(l+1)>>1)
%o A383666     m, k = n, f(n)
%o A383666     while m != k: m, k = k, f(k)
%o A383666     return m # _Chai Wah Wu_, May 21 2025
%Y A383666 Cf. A030130, A158581, A383667.
%K A383666 nonn,base,easy
%O A383666 1,1
%A A383666 _Clark Kimberling_, May 07 2025