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

3, 7, 9, 10, 12, 15, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86

Offset: 1

Author

Clark Kimberling, May 07 2025

Keywords

• nonn
• base
• easy

Comments

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

Examples

			From _David A. Corneth_, May 17 2025: (Start)
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.
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)
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A030130, A158581, A383667.

Programs

Maple

filter:= proc(n) local L,n1,n0;
  L:= convert(n,base,2);
  n1:= convert(L,`+`);
  n0:= nops(L)-n1;
  n1 >= 2 and n0 <> 1
end proc;
select(filter, [$1..1000]); # Robert Israel, May 13 2025

Mathematica

s = Select[Range[200], DigitCount[#, 2, 0] != 1 && DigitCount[#, 2, 1] != 1 &]
Map[First, RealDigits[s, 2]]

PARI

isok(k) = my(b=binary(k)); (#select(x->(x==1), b) != 1) && (#select(x->(x==0), b) != 1); \\ Michel Marcus, May 13 2025

PARI

is(n) = {
    my(v = valuation(n, 2));
    if(n >> v == 1, return(0));
    if(1<> valuation(c, 2) == 1, return(0));
    1
} \\ David A. Corneth, May 17 2025

PARI

upto(n) = {
    my(res = [1..n], del = List());
    for(i = 0, logint(n, 2)+1,
        pow2 = 1<David A. Corneth, May 17 2025

Python

def A383666(n):
    def f(x):
        if x<=1: return n+x
        l, s = x.bit_length(), bin(x)[2:]
        if (m:=s.count('0'))>0: return n+s.index('0')-(m>1)+(l*(l-1)>>1)
        return n-1+(l*(l+1)>>1)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, May 21 2025