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A253317 Indices in A261283 where records occur.

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