A244415 Exponent of 4 appearing in the 4-adic value of 1/n, n >= 1, given in A240226(n).

0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 1

Offset: 1

Author

Wolfdieter Lang, Jun 28 2014

Keywords

• nonn
• easy

Comments

See the comment under A240226 for g-adic value of x and the Mahler reference, p. 7, where this exponent is called f.

Note that the exponent used in the g-adic value of 1/n is also called g-adic valuation of n if g is prime. See e.g., A007814 (g=2) and A007949 (g=3) and the corresponding A006519 and A038500 for the 2-adic and 3-adic value of 1/n, respectively.

Examples

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

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..65537

Crossrefs

Cf. A240226, A007814 (case g=2), A007949 (case g=3).

Programs

Mathematica

Array[IntegerExponent[2 #, 4] &, 105] (* Michael De Vlieger, Nov 06 2018 *)

PARI

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

Python

def A244415(n): return (~n&n-1).bit_length()+1>>1 # Chai Wah Wu, Jul 09 2023

Formula

a(n) = 0 if n is odd, and if n is even a(n) = f(1/n) with f(1/n) the smallest positive integer such that the highest power of 2 in n (that is A006519(n)) divides 4^f(1/n).

a(n) = valuation(2*n, 4). - Andrew Howroyd, Jul 31 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2/3. - Amiram Eldar, Jun 30 2023