A371908 a(n) = 2-adic valuation of A008336(2*n).

0, 1, 3, 2, 5, 4, 2, 1, 5, 4, 2, 1, 4, 3, 1, 0, 5, 4, 2, 1, 4, 3, 1, 0, 4, 3, 1, 0, 3, 2, 0, 1, 7, 6, 4, 3, 0, 1, 3, 2, 6, 5, 3, 2, 5, 4, 2, 1, 6, 5, 3, 4, 1, 0, 2, 1, 5, 4, 2, 1, 4, 3, 1, 0, 7, 6, 4, 3, 0, 1, 3, 2, 6, 5, 3, 2, 5, 4, 2, 1, 6, 5, 3, 2, 5, 4, 2

Offset: 1

Author

Michael De Vlieger, Apr 11 2024

Keywords

• nonn
• easy

Comments

Aside from initial 0, first 50 terms agree with A371905: A371905(50) = 3 while a(51) = 5.

Examples

			Let b(n) = A008336(n) and let f(x) = A007814(x).
a(1) = 0 since b(2*1) = 1 and f(b(2)) = 0.
a(2) = 1 since b(2*2) = 6 and f(b(4)) = 1.
a(3) = 3 since b(2*3) = 120 and f(b(6)) = 3, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007814, A008336, A371905, A372095, A372840.

Programs

Mathematica

k = 1; nn = 240; p[_] := 0; r = 0; q = Prime[k];
{0}~Join~Reap[
    Do[If[AnyTrue[#, p[#1] < #2 & @@ # &],
        Map[p[#1] += #2 & @@ # &, #],
        Map[p[#1] -= #2 & @@ # &, #] ] &@
        Map[{PrimePi[#1], #2} & @@ # &, FactorInteger[n]];
      If[Divisible[n, q], Sow[p[k] ] ], {n, nn}] ][[-1, 1]]

Python

from itertools import count, islice
def A371908_gen(): # generator of terms
    m = 1
    for n in count(1,2):
        a, b = divmod(m,n)
        m = m*n if b else a
        yield (~m&m-1).bit_length()
        a, b = divmod(m,n+1)
        m = m*(n+1) if b else a
A371908_list = list(islice(A371908_gen(),20)) # Chai Wah Wu, Apr 15 2024

Formula

a(n) = A007814(A008336(2*n)).