A130250 Minimal index k of a Jacobsthal number such that A001045(k) >= n (the 'upper' Jacobsthal inverse).

0, 1, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 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, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9

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=2 (see A130249 for another version). a(n+1) is equal to the partial sum of the Jacobsthal indicator sequence (see A105348).

Examples

			a(10)=5 because A001045(5) = 11 >= 10, but A001045(4) = 5 < 10.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

For partial sums see A130252.

Cf. A001045, A105348, A130234, A130242, A130249, A130253.

Programs

Magma

[0] cat [Ceiling(Log(2,3*n-1)): n in [1..120]]; // G. C. Greubel, Mar 18 2023

Mathematica

Table[If[n==0, 0, Ceiling[Log[2, 3*n-1]]], {n,0,120}] (* G. C. Greubel, Mar 18 2023 *)

Python

def A130250(n): return (3*n-2).bit_length() if n else 0 # Chai Wah Wu, Apr 17 2025

SageMath

def A130250(n): return 0 if (n==0) else ceil(log(3*n-1, 2))
[A130250(n) for n in range(121)] # G. C. Greubel, Mar 18 2023

Formula

a(n) = ceiling(log_2(3n-1)) = 1 + floor(log_2(3n-2)) for n >= 1.

a(n) = A130249(n-1) + 1 = A130253(n-1) for n >= 1.

G.f.: (x/(1-x))*Sum_{k>=0} x^A001045(k).