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A135517 a(n) = 2^(A091090(n)-1). 4

%I

%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/pipermail/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-Fran├žois 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

%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

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Last modified May 31 16:29 EDT 2020. Contains 334748 sequences. (Running on oeis4.)