%I #43 May 23 2024 00:49:12
%S 0,1,2,3,8,9,10,11,128,129,130,131,136,137,138,139,32768,32769,32770,
%T 32771,32776,32777,32778,32779,32896,32897,32898,32899,32904,32905,
%U 32906,32907,2147483648,2147483649,2147483650,2147483651,2147483656,2147483657
%N Indices in A261283 where records occur.
%C From _Gus Wiseman_, Dec 29 2023: (Start)
%C 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:
%C 0: 0 ~ {}
%C 1: 1 ~ {1}
%C 2: 10 ~ {2}
%C 3: 11 ~ {1,2}
%C 8: 1000 ~ {4}
%C 9: 1001 ~ {1,4}
%C 10: 1010 ~ {2,4}
%C 11: 1011 ~ {1,2,4}
%C 128: 10000000 ~ {8}
%C 129: 10000001 ~ {1,8}
%C 130: 10000010 ~ {2,8}
%C 131: 10000011 ~ {1,2,8}
%C 136: 10001000 ~ {4,8}
%C 137: 10001001 ~ {1,4,8}
%C 138: 10001010 ~ {2,4,8}
%C 139: 10001011 ~ {1,2,4,8}
%C For powers of 3 we have A368531.
%C (End)
%H Michael De Vlieger, <a href="/A253317/b253317.txt">Table of n, a(n) for n = 1..4096</a>
%H Lorenzo Sauras-Altuzarra, <a href="https://arxiv.org/abs/2002.03075">Some arithmetical problems that are obtained by analyzing proofs and infinite graphs</a>, arXiv:2002.03075 [math.NT], 2020.
%F a(1) = 0 and a(n) = a(n-A053644(n-1)) + 2^(A053644(n-1)-1). - _Lorenzo Sauras Altuzarra_, Dec 18 2019
%F a(n) = A358126(n-1) / 2. - _Tilman Piesk_, Dec 18 2022
%F a(2^n+1) = 2^(2^n-1) = A058891(n+1). - _Gus Wiseman_, Dec 29 2023
%F a(2^n) = A072639(n). - _Gus Wiseman_, Dec 29 2023
%F 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
%p a := proc(n) local k, A:
%p A := [seq(0,i=1..n)]: A[1]:=0:
%p for k from 1 to n-1 do
%p A[k+1] := A[k-2^ilog2(k)+1]+2^(2^ilog2(k)-1): od:
%p return A[n]: end proc: # _Lorenzo Sauras Altuzarra_, Dec 18 2019
%p # second Maple program:
%p a:= n-> (l-> add(l[i+1]*2^(2^i-1), i=0..nops(l)-1))(Bits[Split](n-1)):
%p seq(a(n), n=1..38); # _Alois P. Heinz_, Dec 13 2023
%t Nest[Append[#1, #1[[-#2]] + 2^(#2 - 1)] & @@ {#, 2^(IntegerLength[Length[#], 2] - 1)} &, {0, 1}, 36] (* _Michael De Vlieger_, May 08 2020 *)
%o (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
%Y Cf. A053644 (most significant bit).
%Y A048793 lists binary indices, length A000120, sum A029931.
%Y A070939 gives length of binary expansion.
%Y A096111 gives product of binary indices.
%Y Cf. A058891, A062050, A072639, A326031, A326675, A326702, A367771, A367912, A368183, A368109, A368531.
%K nonn,base
%O 1,3
%A _Philippe Beaudoin_, Dec 30 2014
%E Corrected reference in name from A253315 to A261283. - _Tilman Piesk_, Dec 18 2022