A354538 a(n) is the least k such that A322523(k) = n.

1, 3, 8, 17, 44, 125, 368, 1097, 3284, 9845, 29528, 88577, 265724, 797165, 2391488, 7174457, 21523364, 64570085, 193710248, 581130737, 1743392204

Offset

0,2

Comments

Linked to its derivation from A322523, this sequence has a natural description: "Given a series of boxes (starting at 0), put each positive integer, in sequence, in the first box in which it is not the sum of two different numbers already in that box. The sequence is the smallest number in each box."

The sequence and associated partition are closely related to powers of 3.

Links

Table of n, a(n) for n=0..20.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Formula

For n >= 2, a(n) = (3^n + 7)/2.

For n >= 0, the contents of the n-th box are (conjecture):

1) All the positive integers == b(i) mod 3^(n+1) where 1 <= i <= 2^n and {b(i)} is some strictly increasing sequence of 2^n integers such that b(1)=a(n), b(2^n)= 5*3^(n-1).

2) The additional 'misfit' entry 2*3^n.

[For the correctness and the formula and the conjecture see A322523. - Jianing Song, Oct 17 2022]

Example

1, 2 go in box 0 {1,2};
3=1+2 so goes in box 1 {3};
4 goes in box 0 {1,2,4};
5=1+4 so goes in box 1 {3,5};
6=2+4 so goes in box 1 {3,5,6};
7 goes in box 0 {1,2,4,7};
8=7+1=3+5 so goes in box 2 {8};
...
Box 0: {1,2,4,7,10,13,16,19,22,...} = {1} mod 3 U misfit entry 2;
Box 1: {3,5,6,12,14,21,23,...}      = {3,5} mod 9 U misfit entry 6;
Box 2: {{8,9,11,15} mod 27} U {18};
Box 3: {{17,20,24,26,27,29,33,45} mod 81} U {54};
...

Prog

(Python)
from itertools import count
def A354538(n):
    for k in count(1):
        c, m = 0, k
        while not (a:=divmod(m, 3))[1]:
            c += 1
            m = a[0]
        if not (m==2 or m%3==1):
            c += 1
            m = (m+1)//3-2
            while (a:=divmod(m, 3))[1]:
                c += 1
                m = a[0]
        if c == n: return k # Chai Wah Wu, Oct 15 2022
def A354538(n): return (n<<1)+1 if n<2 else 3**n+7>>1 # based on formula, Chai Wah Wu, Oct 15 2022

Crossrefs

Cf. A322523.

Sequence in context: A046994 A058811 A101822 * A088589 A319764 A063597

Adjacent sequences: A354535 A354536 A354537 * A354539 A354540 A354541

Keyword

nonn,easy

Author

Hugh Williamson, Aug 17 2022

Extensions

a(7)-a(15) from Michel Marcus, Sep 23 2022

a(16)-a(20) from Chai Wah Wu, Oct 15 2022

Status

approved