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

Author

Philippe Beaudoin, Dec 30 2014

Keywords

• nonn
• base

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.

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.

Programs

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 *)

PARI

a(n)={if(n<=1, 0, my(t=1<Andrew Howroyd, Dec 20 2019

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

G.f.: 1/(1-x) * Sum_{k>=0} (2^(-1+2^k))*x^2^k/(1+x^2^k). - John Tyler Rascoe, May 22 2024

Extensions

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