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

1,64

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A370256 A347245 A104488 * A318655 A056626 A366123

Adjacent sequences: A244410 A244411 A244412 * A244414 A244415 A244416

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 27 2014

Status

approved