A130249 Maximal index k of a Jacobsthal number such that A001045(k)<=n (the 'lower' Jacobsthal inverse).

0, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

Author

Hieronymus Fischer, May 20 2007

Keywords

• nonn

Comments

Inverse of the Jacobsthal sequence (A001045), nearly, since a(A001045(n))=n except for n=1 (see A130250 for another version). a(n)+1 is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).

Examples

			a(12)=5, since A001045(5)=11<=12, but A001045(6)=21>12.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000

Crossrefs

For partial sums see A130251.

Other related sequences A130250, A130253, A105348, A001045, A130233, A130241.

Cf. A000523, A078008 (runlengths).

Programs

Magma

[Floor(Log(3*n+1)/Log(2)): n in [0..30]]; // G. C. Greubel, Jan 08 2018

Mathematica

Table[Floor[Log[2, 3*n + 1]], {n, 0, 50}] (* G. C. Greubel, Jan 08 2018 *)

PARI

for(n=0, 30, print1(floor(log(3*n+1)/log(2)), ", ")) \\ G. C. Greubel, Jan 08 2018

PARI

a(n) = logint(3*n+1, 2); \\ Ruud H.G. van Tol, May 12 2024

Python

def A130249(n): return (3*n+1).bit_length()-1 # Chai Wah Wu, Jun 08 2022

Formula

a(n) = floor(log_2(3n+1)).

a(n) = A130250(n+1) - 1 = A130253(n) - 1.

G.f.: 1/(1-x)*(Sum_{k>=1} x^A001045(k)).

a(n) = A000523(3*n+1). - Ruud H.G. van Tol, May 12 2024