A368531 Numbers whose binary indices are all powers of 3, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion.

0, 1, 4, 5, 256, 257, 260, 261, 67108864, 67108865, 67108868, 67108869, 67109120, 67109121, 67109124, 67109125, 1208925819614629174706176, 1208925819614629174706177, 1208925819614629174706180, 1208925819614629174706181, 1208925819614629174706432

Offset: 1

Author

Gus Wiseman, Dec 29 2023

Keywords

• nonn

Comments

For powers of 2 instead of 3 we have A253317.

Examples

			The terms together with their binary expansions and binary indices begin:
         0:                           0 ~ {}
         1:                           1 ~ {1}
         4:                         100 ~ {3}
         5:                         101 ~ {1,3}
       256:                   100000000 ~ {9}
       257:                   100000001 ~ {1,9}
       260:                   100000100 ~ {3,9}
       261:                   100000101 ~ {1,3,9}
  67108864: 100000000000000000000000000 ~ {27}
  67108865: 100000000000000000000000001 ~ {1,27}
  67108868: 100000000000000000000000100 ~ {3,27}
  67108869: 100000000000000000000000101 ~ {1,3,27}
  67109120: 100000000000000000100000000 ~ {9,27}
  67109121: 100000000000000000100000001 ~ {1,9,27}
  67109124: 100000000000000000100000100 ~ {3,9,27}
  67109125: 100000000000000000100000101 ~ {1,3,9,27}
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..256

Crossrefs

A000244 lists powers of 3.

A048793 lists binary indices, length A000120, sum A029931.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A058891, A062050, A072639, A253317, A326031, A326675, A326702, A367912, A368183, A368109.

Programs

Mathematica

Select[Range[0,10000],IntegerQ[Log[3,Times@@Join@@Position[Reverse[IntegerDigits[#,2]],1]]]&]
(* Second program *)
{0}~Join~Array[FromDigits[Reverse@ ReplacePart[ConstantArray[0, Max[#]], Map[# -> 1 &, #]], 2] &[3^(Position[Reverse@ IntegerDigits[#, 2], 1][[;; , 1]] - 1)] &, 255] (* Michael De Vlieger, Dec 29 2023 *)

Formula

a(3^n) = 2^(3^n - 1).