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

1,8

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.

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

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

Example

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).

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A157187 A152140 A292252 * A104975 A191254 A106404

Adjacent sequences: A244412 A244413 A244414 * A244416 A244417 A244418

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 28 2014

Status

approved