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A238462 2-adic valuation of A052129.

%I A238462 #42 Oct 08 2021 07:31:30
%S A238462 0,0,1,2,6,12,25,50,103,206,413,826,1654,3308,6617,13234,26472,52944,
%T A238462 105889,211778,423558,847116,1694233,3388466,6776935,13553870,
%U A238462 27107741,54215482,108430966,216861932,433723865,867447730
%N A238462 2-adic valuation of A052129.
%H A238462 Kenny Lau, <a href="/A238462/b238462.txt">Table of n, a(n) for n = 0..3323</a>
%F A238462 From _Ridouane Oudra_, Sep 03 2019: (Start)
%F A238462 a(n) = Sum_{i=1..n} 2^(n-i)*v_2(i), where v_2(i) = A007814(i).
%F A238462 More generally, the p-adic valuation of A052129 for any prime p is given by
%F A238462 v_p(A052129(n)) = Sum_{i=1..n} 2^(n-i)*v_p(i), where v_p(i) is the exponent of the highest power of p dividing i. (End)
%p A238462 with(padic): seq(add(2^(n-i)*ordp(i, 2), i=1..n), n=0..60); # _Ridouane Oudra_, Sep 03 2019
%t A238462 Map[IntegerExponent[#, 2] &, Nest[Append[#, Length[#]*#[[-1]]^2] &, {1}, 31]] (* or, per first formula, more efficiently, *)
%t A238462 Array[Sum[2^(# - i)*IntegerExponent[i, 2], {i, #}] &, 32, 0] (* _Michael De Vlieger_, Sep 29 2019 *)
%o A238462 (PARI)
%o A238462 A052129(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^2^k));
%o A238462 a(n) = valuation(A052129(n),2);
%o A238462 (PARI) a(n) = fromdigits(vector(n,i,valuation(i,2)), 2); \\ _Kevin Ryde_, Oct 08 2021
%o A238462 (Python)
%o A238462 n=4000; val=[0]*(n+1); exp=2
%o A238462 while exp <= n:
%o A238462     for j in range(exp,n+1,exp): val[j] += 1
%o A238462     exp *= 2
%o A238462 res = 0; i = 0
%o A238462 while len(str(res)) <= 1000: print(i,res); i += 1; res = res * 2 + val[i]
%o A238462 # _Kenny Lau_, Jun 09 2018
%K A238462 nonn
%O A238462 0,4
%A A238462 _Joerg Arndt_, Feb 27 2014
%E A238462 Incorrect comment removed by _Michel Marcus_, Oct 06 2019