A371093 a(n) is the 2-adic valuation of 3n+1.

0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 6, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 5, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 8, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2

Offset: 0

Author

Antti Karttunen, Apr 19 2024

Keywords

• nonn
• easy

Comments

When a(n) is applied to square array A257852 we obtain square array A004736, or in other words, a(n) applied to any odd number gives the index of the row where it is located in array A257852.

See further comments in A087230.

The asymptotic density of the occurrences of k = 0, 1, 2, ... is 1/2^(k+1). The asymptotic mean of this sequence is 1. - Amiram Eldar, May 28 2024

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537

Crossrefs

Bisections: A000004, A087230.

Cf. A004736, A007814, A016777, A067745, A257852.

Cf. also A371092.

Programs

Mathematica

Table[IntegerExponent[3*n+1, 2], {n, 0, 105}] (* James C. McMahon, Apr 21 2024 *)

PARI

A371093(n) = valuation(1+3*n,2);

Python

def A371093(n): return ((m:=3*n) & ~(m+1)).bit_length() # Chai Wah Wu, Apr 20 2024

Formula

a(n) = A007814(A016777(n)).

For all n >= 0, A067745(1+n) = A016777(n) / 2^a(n).

G.f.: Sum_{k>=1} k*x^(-1/3 + (-2)^(k + 1)/3 + 2^k)/(1 - x^(2^(k + 1))). - Miles Wilson, Sep 30 2024