A253317 Indices in A261283 where records occur.

0, 1, 2, 3, 8, 9, 10, 11, 128, 129, 130, 131, 136, 137, 138, 139, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906, 32907, 2147483648, 2147483649, 2147483650, 2147483651, 2147483656, 2147483657

Offset

1,3

Comments

From Gus Wiseman, Dec 29 2023: (Start)

These are numbers whose binary indices are all powers of 2, where a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, the terms together with their binary expansions and binary indices begin:

0: 0 ~ {}

1: 1 ~ {1}

2: 10 ~ {2}

3: 11 ~ {1,2}

8: 1000 ~ {4}

9: 1001 ~ {1,4}

10: 1010 ~ {2,4}

11: 1011 ~ {1,2,4}

128: 10000000 ~ {8}

129: 10000001 ~ {1,8}

130: 10000010 ~ {2,8}

131: 10000011 ~ {1,2,8}

136: 10001000 ~ {4,8}

137: 10001001 ~ {1,4,8}

138: 10001010 ~ {2,4,8}

139: 10001011 ~ {1,2,4,8}

For powers of 3 we have A368531.

(End)

Links

Michael De Vlieger, Table of n, a(n) for n = 1..4096

Lorenzo Sauras-Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.

Formula

a(1) = 0 and a(n) = a(n-A053644(n-1)) + 2^(A053644(n-1)-1). - Lorenzo Sauras Altuzarra, Dec 18 2019

a(n) = A358126(n-1) / 2. - Tilman Piesk, Dec 18 2022

a(2^n+1) = 2^(2^n-1) = A058891(n+1). - Gus Wiseman, Dec 29 2023

a(2^n) = A072639(n). - Gus Wiseman, Dec 29 2023

Maple

a := proc(n) local k, A:
A := [seq(0, i=1..n)]: A[1]:=0:
for k from 1 to n-1 do
   A[k+1] := A[k-2^ilog2(k)+1]+2^(2^ilog2(k)-1): od:
return A[n]: end proc: # Lorenzo Sauras Altuzarra, Dec 18 2019
# second Maple program:
a:= n-> (l-> add(l[i+1]*2^(2^i-1), i=0..nops(l)-1))(Bits[Split](n-1)):
seq(a(n), n=1..38);  # Alois P. Heinz, Dec 13 2023

Mathematica

Nest[Append[#1, #1[[-#2]] + 2^(#2 - 1)] & @@ {#, 2^(IntegerLength[Length[#], 2] - 1)} &, {0, 1}, 36] (* Michael De Vlieger, May 08 2020 *)

Prog

(PARI) a(n)={if(n<=1, 0, my(t=1<<logint(n-1, 2)); a(n-t) + 2^(t-1))} \\ Andrew Howroyd, Dec 20 2019

Crossrefs

Cf. A053644 (most significant bit).

A048793 lists binary indices, length A000120, sum A029931.

A070939 gives length of binary expansion.

A096111 gives product of binary indices.

Cf. A058891, A062050, A072639, A326031, A326675, A326702, A367771, A367912, A368183, A368109, A368531.

Sequence in context: A037462 A309125 A037318 * A277971 A080288 A278560

Adjacent sequences: A253314 A253315 A253316 * A253318 A253319 A253320

Keyword

nonn,base

Author

Philippe Beaudoin, Dec 30 2014

Extensions

Corrected reference in name from A253315 to A261283. - Tilman Piesk, Dec 18 2022

Status

approved