# [seqfan] Re: An Integer Sequence Game

Veikko Pohjola veikko at nordem.fi
Sun Feb 21 16:58:00 CET 2010

```Leroy,
I guess you did not look for a general solution as a function of n :-) So I
let n=8. The sequence for the maximum score is r(k) = {4,5,6,3,1,2,7,8},
from which m(k) = {4,21,128,387,391,787,5514,44118} yielding a(k) =
{1,1,4,1,1,1,1,387} summing up to 397.
Best regards,
Veikko Pohjola

----- Original Message -----
From: "Leroy Quet" <q1qq2qqq3qqqq at yahoo.com>
To: <seqfan at seqfan.eu>
Sent: Tuesday, February 16, 2010 6:57 PM
Subject: [seqfan] An Integer Sequence Game

> (I cross-posted this to sci.math and rec.puzzles, and to my games blog at:
> http://gamesconceived.blogspot.com/2010/02/integer-sequence-game.html )
>
> The question I am asking is at the bottom.
>
> ---
>
> This is a game for any number of players.
>
> Needed: Pencil/pen, paper, calculator (with long display) perhaps. (Maybe
> this game could be played via a computer running the appropriate program.)
>
> Start by writing down the integers 1, 2, 3,..., n, where n is at least 8
> or more if the number of players is 2, I suggest. n is larger if there are
> more than 2 players.
> This list of integers is called the "r-list".
>
> The variable m starts the game with the value 1. In other words, m(0) = 1.
>
> Players take turns. On the kth move (the kth move among all players
> together), the moving player lets r(k) = any uncircled integer from the
> r-list.
> The player then circles that number.
>
> m(k) is the value of m after the kth move.
> Let m(k) =
> r(k)*m(k-1) + (number of composites among m(0),m(1),m(2),...,m(k-1)).
>
> Add to the moving player's score the largest value from
> m(0),m(1),m(2),...m(k-1) that divides m(k).
>
> The move is complete when the moving player writes down m(k) at the end of
> the growing list of the values of m.
>
> Players keep taking turns until k = n.
>
> ---
>
> Example game, n = 8: (I may have made a mistake with my math.)
> m(0) = 1
> r(1) = 2; m(1) = 2*1+0 = 2. (Prime.)
> Moving player gets 1 added to score.
> r(2) = 8; m(2) = 8*2+0 = 16. (Composite.)
> Moving player gets 2 added to score.
> r(3) = 3; m(3) = 16*3+1 = 49. (Composite.)
> Moving player gets 1 added to score.
> r(4) = 5; m(4) = 49*5+2 = 247. (Composite.)
> Moving player gets 1 added to score.
> r(5) = 1; m(5) = 247*1+3 = 250. (Composite.)
> Moving player gets 2 added to score.
> r(6) = 4; m(6) = 250*4+4 = 1004. (Composite.)
> Moving player gets 2 added to score.
> r(7) = 6; m(7) = 1004*6+5 = 6029. (Prime)
> Moving player gets 1 added to score.
> r(8) = 7; m(8) = 6029*7+5 = 42208. (Composite, but this does not matter.)
> Moving player gets 16 added to score.
>
> ---
>
> How does the sequence {a(k)} begin, letting a(n) = the largest possible
> score for a 1-person game where the r-list contains the first n positive
> integers?
>
> Thanks,
> Leroy Quet
>
>
> [ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] ) ]
>
>
>
>
>
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>

```