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%I #50 Jan 02 2024 21:49:29
%S 1,4,3,12,5,12,7,32,9,20,11,36,13,28,15,80,17,36,19,60,21,44,23,96,25,
%T 52,27,84,29,60,31,192,33,68,35,108,37,76,39,160,41,84,43,132,45,92,
%U 47,240,49,100,51,156,53,108,55,224,57,116,59,180,61,124,63
%N a(n) is the largest integer m such that 2^m divides (2*n)^n, i.e., the exponent of 2 in (2*n)^n.
%C n times one more than the trailing 0's in the binary representation of n. - _Ralf Stephan_, Aug 22 2013
%H Vincenzo Librandi, <a href="/A091512/b091512.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.
%F a(n) = A007814(A000312(n)) = n*A001511(n) = A069895(n)/2.
%F G.f.: Sum_{k>=0} 2^k*x^2^k/(1-x^2^k)^2.
%F Recurrence: a(0) = 0, a(2*n) = 2*a(n) + 2*n, a(2*n+1) = 2*n+1.
%F Dirichlet g.f.: zeta(s-1)*2^s/(2^s-2). - _Ralf Stephan_, Jun 17 2007
%F Mobius transform of A162728, where x/(1-x)^2 = Sum_{n>=1} A162728(n)*x^n/(1+x^n). - _Paul D. Hanna_, Jul 12 2009
%F a(n) = A162728(2*n)/phi(2*n), where x/(1-x)^2 = Sum_{n>=1} A162728(n)*x^n/(1+x^n). - _Paul D. Hanna_, Jul 12 2009
%F a((2*n-1)*2^p) = (2*n-1)*(p+1)*2^p, p >= 0. Observe that a(2^p) = A001787(p+1). - _Johannes W. Meijer_, Feb 08 2013
%F Sum_{k=1..n} a(k) ~ n^2. - _Amiram Eldar_, Oct 22 2022
%F a(n) = Sum_{d divides n} d*A048298(n/d); that is, a(n) is the Dirichlet product of A048298(n) and A000027(n). - _Peter Bala_, Jan 02 2024
%p nmax:=63: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (2*n-1)*(p+1)*2^p od: od: seq(a(n), n=1..nmax); # _Johannes W. Meijer_, Feb 08 2013
%p # second Maple program:
%p a:= n-> n*padic[ordp](2*n, 2):
%p seq(a(n), n=1..63); # _Alois P. Heinz_, Jan 02 2024
%t Table[ Part[ Flatten[ FactorInteger[(2 n)^n]], 2], {n, 1, 124}]
%t Table[IntegerExponent[(2n)^n,2],{n,70}] (* _Harvey P. Dale_, Sep 11 2015 *)
%o (PARI) a(n)=n*(valuation(n,2)+1)
%o (PARI) a(n)=if(n<1,0,if(n%2==0,2*a(n/2)+n,n))
%o (Magma) [n*(Valuation(n, 2)+1): n in [1..80]]; // _Vincenzo Librandi_, May 16 2013
%o (Python)
%o def A091512(n): return n*(n&-n).bit_length() # _Chai Wah Wu_, Jul 11 2022
%Y Cf. A091519, A090740, A090739, A048298, A162728, A220466.
%K nonn,mult,easy
%O 1,2
%A _Ralf Stephan_ and _Labos Elemer_, Jan 18 2004
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