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A244415
Exponent of 4 appearing in the 4-adic value of 1/n, n >= 1, given in A240226(n).
5
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
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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jun 28 2014
STATUS
approved