A307096 Positive integers m such that for any positive integer k the last k bits of the binary expansion of m is not a multiple of 3.

1, 5, 13, 17, 29, 37, 49, 61, 65, 77, 101, 113, 125, 133, 145, 157, 193, 205, 229, 241, 253, 257, 269, 293, 305, 317, 389, 401, 413, 449, 461, 485, 497, 509, 517, 529, 541, 577, 589, 613, 625, 637, 769, 781, 805, 817, 829, 901, 913, 925, 961, 973, 997, 1009

Offset

1,2

Comments

The number of terms less than 2^n is the n-th Fibonacci number F(n), A000045.

The number of terms between 2^(n-1) and 2^n in the sequence is the Fibonacci number F(n-2), A000045.

If 2^(n-1) <= x < 2^n, then x is in the sequence if and only if x is not divisible by 3 and x - 2^(n-1) is in the sequence. - Robert Israel, Apr 25 2019

Links

Table of n, a(n) for n=1..54.

Formula

(a(n)+1)/2 = A219608(n), the n-th odd term in A060142.

Example

29 is 11101_2 and none of 11101_2, 1101_2, 101_2, 1_2 are divisible by 3.

Maple

f := n-> if(n != 0, add(2^(k-1)*`if`((n mod 2^k) mod 3 = 0, 1, 0), k = 1 .. ceil(log(n)/log(2))), 0);
ker := []; for n from 1 to 1024 do if f(n) = 0 then ker := [op(ker), n] end if end do; ker;
# Alternative:
A1:= {1}: A2:= {}:
for d from 1 to 12 do
  if d::odd then A1:= A1 union map(`+`, A2, 2^d)
  else A2:= A2 union map(`+`, A1, 2^d)
  fi
od:
sort(convert(A1 union A2, list)); # Robert Israel, Apr 25 2019

Mathematica

Select[Range[10^3], Function[s, NoneTrue[Array[FromDigits[Take[s, -#], 2] &, Length@ s], Mod[#, 3] == 0 &]]@ IntegerDigits[#, 2] &] (* Michael De Vlieger, Mar 24 2019 *)

Prog

(PARI) isok(n) = {if (n % 3, my(b=binary(n)); for (k=1, #b-1, b[k] = 0; if ((fromdigits(b, 2) % 3) == 0, return (0)); ); return (1); ); return (0); } \\ Michel Marcus, Apr 24 2019

Crossrefs

Cf. A060142, A219608, A000045.

Sequence in context: A349900 A347163 A111055 * A283391 A145016 A354155

Adjacent sequences: A307093 A307094 A307095 * A307097 A307098 A307099

Keyword

easy,base,nonn

Author

John Rickert, Mar 24 2019

Status

approved