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a(n) = A007814(n) + A036987(n-1) + 1.
6

%I #15 Jul 07 2022 02:02:36

%S 2,3,1,4,1,2,1,5,1,2,1,3,1,2,1,6,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,7,1,2,

%T 1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,8,1,2,1,3,

%U 1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,1,2,1,4,1

%N a(n) = A007814(n) + A036987(n-1) + 1.

%H Reinhard Zumkeller, <a href="/A135560/b135560.txt">Table of n, a(n) for n = 1..10000</a>

%F a(2^k) = k+2; a(2^k + 2^(k-1)) = k. - _Reinhard Zumkeller_, Mar 02 2008

%t a[n_] := 1 + IntegerExponent[n, 2] + Sum[(-1)^(n - k - 1)*Binomial[n - 1, k]* Sum[Binomial[k, 2^j - 1], {j, 0, k}], {k, 0, n - 1}]; Table[a[n], {n, 1, 25}] (* _G. C. Greubel_, Oct 17 2016 *)

%o (PARI) a(n)=my(t=valuation(n, 2)); t + (n==2^t) + 1 \\ _Charles R Greathouse IV_, Oct 17 2016

%o (Python)

%o def A135560(n): return (m:=(~n & n-1)).bit_length()+int(m==n-1)+1 # _Chai Wah Wu_, Jul 06 2022

%Y Cf. A007814, A036987, A091090, A135534, A135561.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_, Mar 01 2008

%E More terms from _Reinhard Zumkeller_, Mar 02 2008