A238462 - OEIS

%I #42 Oct 08 2021 07:31:30

%S 0,0,1,2,6,12,25,50,103,206,413,826,1654,3308,6617,13234,26472,52944,

%T 105889,211778,423558,847116,1694233,3388466,6776935,13553870,

%U 27107741,54215482,108430966,216861932,433723865,867447730

%N 2-adic valuation of A052129.

%H Kenny Lau, <a href="/A238462/b238462.txt">Table of n, a(n) for n = 0..3323</a>

%F From _Ridouane Oudra_, Sep 03 2019: (Start)

%F a(n) = Sum_{i=1..n} 2^(n-i)*v_2(i), where v_2(i) = A007814(i).

%F More generally, the p-adic valuation of A052129 for any prime p is given by

%F 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 with(padic): seq(add(2^(n-i)*ordp(i, 2), i=1..n), n=0..60); # _Ridouane Oudra_, Sep 03 2019

%t Map[IntegerExponent[#, 2] &, Nest[Append[#, Length[#]*#[[-1]]^2] &, {1}, 31]] (* or, per first formula, more efficiently, *)

%t Array[Sum[2^(# - i)*IntegerExponent[i, 2], {i, #}] &, 32, 0] (* _Michael De Vlieger_, Sep 29 2019 *)

%o (PARI)

%o A052129(n) = if( n<1, n==0, prod(k=0, n-1, (n - k)^2^k));

%o a(n) = valuation(A052129(n),2);

%o (PARI) a(n) = fromdigits(vector(n,i,valuation(i,2)), 2); \\ _Kevin Ryde_, Oct 08 2021

%o (Python)

%o n=4000; val=[0]*(n+1); exp=2

%o while exp <= n:

%o     for j in range(exp,n+1,exp): val[j] += 1

%o     exp *= 2

%o res = 0; i = 0

%o while len(str(res)) <= 1000: print(i,res); i += 1; res = res * 2 + val[i]

%o # _Kenny Lau_, Jun 09 2018

%K nonn

%O 0,4

%A _Joerg Arndt_, Feb 27 2014

%E Incorrect comment removed by _Michel Marcus_, Oct 06 2019