%I #36 Sep 18 2024 09:23:37
%S 1,1,1,2,1,2,1,4,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,
%T 1,4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,1,
%U 4,1,2,1,8,1,2,1,4,1,2,1,16,1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,32,1,2,1,4,1,2,1,8
%N a(n) = 2^(A091090(n)-1).
%C Also, a(n) = denominator(Euler(n, x) - Euler(n, 1)). - Observation from _Peter Luschny_, Aug 08 2017, proof from _Vladimir Shevelev_, Aug 13 2017
%C Also, a(n) = denominator(Euler(n,x) + Euler(n,0)). - _Vladimir Shevelev_, Aug 09 2017
%H Robert Israel, <a href="/A135517/b135517.txt">Table of n, a(n) for n = 0..10000</a>
%H Vladimir Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-August/017868.html">Is A290646 = A135517?</a>, Posting to Sequence Fans Mailing List, Aug 13 2017
%H Vladimir Shevelev, <a href="https://arxiv.org/abs/1708.08096">On a Luschny question</a>, arXiv:1708.08096 [math.NT], 2017.
%F For n >= 1, a(n) = 2^max_{odd k=1..n} (A007814(k+1) - t(n,k) - delta(n,k)), where delta(n,k) is the Kronecker symbol: delta(i,j) is 1 if i=j and 0 otherwise, and t(n,k) is the number of carries which appear in the addition of k and n-k in base 2. This allows us to answer in the affirmative the author's question (for a proof see Shevelev's link and its continuations). - _Vladimir Shevelev_, Aug 15 2017
%p GS(2,5,200); # see A135416.
%p a := n -> `if`(n=1 or n mod 2 = 0, 1, 2*a(iquo(n,2))):
%p seq(a(n), n=0..103); # _Peter Luschny_, Aug 09 2017
%t b[n_] := b[n] = Which[n==0, 1, n==1, 1, EvenQ[n], 1, True, b[(n-1)/2] + 1]; a[n_] := 2^(b[n+1]-1); Array[a,103,0] (* _Jean-François Alcover_, Aug 12 2017 *)
%o (PARI) a(n)=my(m=valuation(n+1,2)); 2^if(n>>m, m, m-1) \\ _Charles R Greathouse IV_, Aug 15 2017
%o (Python)
%o def A135517(n): return (1<<(~(n+1)&n).bit_length()-(not n&(n+1))) if n else 1 # _Chai Wah Wu_, Sep 18 2024
%Y This is Guy Steele's sequence GS(2, 5) (see A135416).
%Y Cf. A091090.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, based on a message from Guy Steele and _Don Knuth_, Mar 01 2008
%E Entry revised by _N. J. A. Sloane_, Aug 31 2017