A244413 Exponent of highest power of 8 dividing n.

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

Offset: 1

Author

Wolfdieter Lang, Jun 27 2014

Keywords

• nonn
• easy

Comments

This is the member g = 8 in the g-family of sequences, g integer >= 2, call it phi(g,n), n >= 1. In the Mahler reference, Lemma 2, pp. 6-7, this exponent is called f = -phi if g divides r = n (s = 1 there), and f = 0 if g does not divide r = n (s = 1 there).

References

• Kurt Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65536

Crossrefs

Cf. A007814, A007949, A235127, A112765, A122841, A214411.

Programs

Mathematica

Table[IntegerExponent[n, 8], {n, 1, 100}] (* Amiram Eldar, Sep 14 2020 *)

PARI

A244413(n) = valuation(n,8); \\ Antti Karttunen, Oct 07 2017

Python

def A244413(n): return (~n&n-1).bit_length()//3 # Chai Wah Wu, Jul 09 2022

Formula

n = 8^a(n)*m with a(n) nonnegative integer such that 8 does not divide m, for n >= 1.

O.g.f.: Sum_{k>=1} x^(8^k)/(1-x^(8^k)).

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/7. - Amiram Eldar, Jan 17 2022

a(n) = floor(A007814(n)/3). - Alan Michael Gómez Calderón, Jul 25 2024