A240226 4-adic value of 1/n, n >= 1.

1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 64, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 64, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4, 1, 4, 1, 4, 1, 16, 1, 4

Offset: 1

Author

Wolfdieter Lang, Jun 28 2014

Keywords

• nonn
• easy
• mult

Comments

For the definition of g-adic value of x, called |x|_g with g an integer >= 2, see the Mahler reference, p. 7. Sometimes also called g-adic absolute value of x. If g is not a prime then this is called a non-archimedean pseudo-valuation. See Mahler, p. 10.

Examples

			n = 2: A006519(2) = 1, 2 divides 4^1, hence f(1/2) = 1 and a(2) = 4^1 = 4.
n = 4: A006519(4) = 2^2, 4 divides 4^1, hence f(1/4) = 1 and a(4) = 4.
n = 8: A006519(8) = 2^3, 8 does not divide 4^1 but 4^2, hence f(1/8) = 2 and a(8) = 4^2 = 16.
		

References

• Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16383

Crossrefs

Cf. A001620, A006519, A038500 (3-adic value of 1/n), A244415.

Programs

Mathematica

Array[4^IntegerExponent[2 #, 4] &, 90] (* Michael De Vlieger, Nov 06 2018 *)

PARI

a(n) = 4^ceil(valuation(n, 2)/2); \\ Andrew Howroyd, Jul 31 2018

Python

def A240226(n): return 1<<((~n&n-1).bit_length()+1&-2) # Chai Wah Wu, Jul 09 2023

Formula

a(n) = 1 if n is odd. a(n) = 4^f(1/n) if n is even, where f(1/n) is the smallest positive integer such that the highest power of 2 in n (that is A006519(n)) divides 4^f(1/n). The f(1/n) values are given in A244415(n).

From Andrew Howroyd, Jul 31 2018: (Start)

a(n) = 4^valuation(2*n, 4) = 4^A244415(n).

Multiplicative with a(2^e) = 4^ceiling(e/2), a(p^e) = 1 for odd prime p. (End)

From Amiram Eldar, Oct 24 2023: (Start)

Dirichlet g.f.: zeta(s)*(2^s-1)*(2^s+4)/(4^s-4).

Sum_{k=1..n} a(k) ~ (3/(4*log(2))) * n * (log(n) + gamma + 4*log(2)/3 - 1), where gamma is Euler's constant (A001620). (End)