A238462 2-adic valuation of A052129.

0, 0, 1, 2, 6, 12, 25, 50, 103, 206, 413, 826, 1654, 3308, 6617, 13234, 26472, 52944, 105889, 211778, 423558, 847116, 1694233, 3388466, 6776935, 13553870, 27107741, 54215482, 108430966, 216861932, 433723865, 867447730

Offset: 0

Author

Joerg Arndt, Feb 27 2014

Keywords

• nonn

Links

• Kenny Lau, Table of n, a(n) for n = 0..3323

Programs

Maple

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

Mathematica

Map[IntegerExponent[#, 2] &, Nest[Append[#, Length[#]*#[[-1]]^2] &, {1}, 31]] (* or, per first formula, more efficiently, *)
Array[Sum[2^(# - i)*IntegerExponent[i, 2], {i, #}] &, 32, 0] (* Michael De Vlieger, Sep 29 2019 *)

PARI

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

PARI

a(n) = fromdigits(vector(n,i,valuation(i,2)), 2); \\ Kevin Ryde, Oct 08 2021

Python

n=4000; val=[0]*(n+1); exp=2
while exp <= n:
    for j in range(exp,n+1,exp): val[j] += 1
    exp *= 2
res = 0; i = 0
while len(str(res)) <= 1000: print(i,res); i += 1; res = res * 2 + val[i]
# Kenny Lau, Jun 09 2018

Formula

From Ridouane Oudra, Sep 03 2019: (Start)

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

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

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)

Extensions

Incorrect comment removed by Michel Marcus, Oct 06 2019