A367912 -id:A367912 - cp's OEIS frontend

A368185 Sorted list of positions of first appearances in A368183 (number of sets that can be obtained by choosing a different binary index of each binary index).

1, 4, 7, 20, 276, 320, 1088, 65856, 66112, 66624

Offset: 1

Gus Wiseman, Dec 18 2023

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
      1: {{1}}
      4: {{1,2}}
      7: {{1},{2},{1,2}}
     20: {{1,2},{1,3}}
    276: {{1,2},{1,3},{1,4}}
    320: {{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
  65856: {{1,2,3},{1,4},{1,5}}
  66112: {{1,2,3},{2,4},{1,5}}
  66624: {{1,2,3},{1,2,4},{1,5}}

For sequences we have A367911, unsorted A367910, firsts of A367905.
Multisets w/o distinctness: A367915, unsorted A367913, firsts of A367912.
Sequences w/o distinctness: A368112, unsorted A368111, firsts of A368109.
Sorted list of positions of first appearances in A368183.
The unsorted version is A368184.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A326031, A326702, A326753, A355739, A355741, A367771, A367906, A367907.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]]],{n,1000}];
Select[Range[Length[c]], FreeQ[Take[c,#-1],c[[#]]]&]

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.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

For powers of 2 instead of 3 we have A253317.

			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}

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

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.

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 *)

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

cp's OEIS Frontend

A368185 Sorted list of positions of first appearances in A368183 (number of sets that can be obtained by choosing a different binary index of each binary index).

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