A funny number sequence

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A funny number sequence#508331

By arcis - 12 Nov 2004 19:53

- 12 Nov 2004 19:53 #508331

1. Pick any positive integer n.

2. If n is even, divide it by two;
if it is odd, multiply it by three and add one.

3. If n = 1, stop;
else go back to step 2.

The conjecture says that this process always stops, no matter what the start value.

#508638

By Le Rouge - 13 Nov 2004 02:54

- 13 Nov 2004 02:54 #508638

Odd(3)
3 * 3 +1 = 10

Even(10)
10 / 2

Odd(5)
5 * 3 + 1 = 16

Even(16)
16/2 = 8

Even(8)
8/2 = 4

Even(4)
4/2 = 2

Even(2)
2/2 = 1

Impressive.

#508730

By Cap - 13 Nov 2004 08:30

- 13 Nov 2004 08:30 #508730

When you multiply 2 odd numbers, the result is always odd. Then if you add one to this result, it must always be even. Instead of using 3, the instructions could have said multiply by 5, or 7, or 9, or 11 etc. and then add 1.

And since you will be dividing this inevitable even number by the smallest even number, 2, the consequent results will be even, or when odd they instantly become even again, and being factored by 2 until reaching the number 1.

Interesting.

Cap

+#508876

By arcis - 13 Nov 2004 13:58

- 13 Nov 2004 13:58 #508876

And since you will be dividing this inevitable even number by the smallest even number, 2, the consequent results will be even"

I don't think so. For example: 26 / 2 = 13.

Whenever the result is even, the result of the next step will be smaller than the result of the previous step.
Whenever the result is odd, the result of the next step will be bigger than the result of the previous step.

So, in some way there must be more decreasing steps than increasing steps for each start number.

Here are some examples:

start x: 1

start x: 2
1
start x: 3
5 8 4 2 1
start x: 4
2 1
start x: 5
8 4 2 1
start x: 6
3 5 8 4 2 1
start x: 7
11 17 26 13 20 10 5 8 4 2 1
start x: 8
4 2 1
start x: 9
14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 10
5 8 4 2 1
start x: 11
17 26 13 20 10 5 8 4 2 1
start x: 12
6 3 5 8 4 2 1
start x: 13
20 10 5 8 4 2 1
start x: 14
7 11 17 26 13 20 10 5 8 4 2 1
start x: 15
23 35 53 80 40 20 10 5 8 4 2 1
start x: 16
8 4 2 1
start x: 17
26 13 20 10 5 8 4 2 1
start x: 18
9 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 19
29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 20
10 5 8 4 2 1
start x: 21
32 16 8 4 2 1
start x: 22
11 17 26 13 20 10 5 8 4 2 1
start x: 23
35 53 80 40 20 10 5 8 4 2 1
start x: 24
12 6 3 5 8 4 2 1
start x: 25
38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 26
13 20 10 5 8 4 2 1
start x: 27
41 62 31 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890
445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911
1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 28
14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 29
44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 30
15 23 35 53 80 40 20 10 5 8 4 2 1
start x: 31
47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445 668
334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911 1367
2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53 80 40
20 10 5 8 4 2 1
start x: 32
16 8 4 2 1
start x: 33
50 25 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 34
17 26 13 20 10 5 8 4 2 1
start x: 35
53 80 40 20 10 5 8 4 2 1
start x: 36
18 9 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 37
56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 38
19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 39
59 89 134 67 101 152 76 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 40
20 10 5 8 4 2 1
start x: 41
62 31 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445
668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911
1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 42
21 32 16 8 4 2 1
start x: 43
65 98 49 74 37 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 44
22 11 17 26 13 20 10 5 8 4 2 1
start x: 45
68 34 17 26 13 20 10 5 8 4 2 1
start x: 46
23 35 53 80 40 20 10 5 8 4 2 1
start x: 47
71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911 1367 2051
3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 48
24 12 6 3 5 8 4 2 1
start x: 49
74 37 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 50
25 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 51
77 116 58 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 52
26 13 20 10 5 8 4 2 1
start x: 53
80 40 20 10 5 8 4 2 1
start x: 54
27 41 62 31 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593
890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822
911 1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35
53 80 40 20 10 5 8 4 2 1
start x: 55
83 125 188 94 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593
890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822
911 1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35
53 80 40 20 10 5 8 4 2 1
start x: 56
28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 57
86 43 65 98 49 74 37 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 58
29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 59
89 134 67 101 152 76 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 60
30 15 23 35 53 80 40 20 10 5 8 4 2 1
start x: 61
92 46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 62
31 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445
668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911
1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53
80 40 20 10 5 8 4 2 1
start x: 63
95 143 215 323 485 728 364 182 91 137 206 103 155 233 350 175 263 395 593 890 445
668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911
1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53
80 40 20 10 5 8 4 2 1
start x: 64
32 16 8 4 2 1
start x: 65
98 49 74 37 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 66
33 50 25 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 67
101 152 76 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 68
34 17 26 13 20 10 5 8 4 2 1
start x: 69
104 52 26 13 20 10 5 8 4 2 1
start x: 70
35 53 80 40 20 10 5 8 4 2 1
start x: 71
107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445 668 334
167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911 1367 2051
3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53 80 40 20 10
5 8 4 2 1
start x: 72
36 18 9 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 73
110 55 83 125 188 94 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263
395 593 890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429
3644 1822 911 1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61
92 46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 74
37 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 75
113 170 85 128 64 32 16 8 4 2 1
start x: 76
38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 77
116 58 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 78
39 59 89 134 67 101 152 76 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 79
119 179 269 404 202 101 152 76 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 80
40 20 10 5 8 4 2 1
start x: 81
122 61 92 46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 82
41 62 31 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890
445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911
1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53
80 40 20 10 5 8 4 2 1
start x: 83
125 188 94 47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593
890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822
911 1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35
53 80 40 20 10 5 8 4 2 1
start x: 84
42 21 32 16 8 4 2 1
start x: 85
128 64 32 16 8 4 2 1
start x: 86
43 65 98 49 74 37 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 87
131 197 296 148 74 37 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 88
44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 89
134 67 101 152 76 38 19 29 44 22 11 17 26 13 20 10 5 8 4 2 1
start x: 90
45 68 34 17 26 13 20 10 5 8 4 2 1
start x: 91
137 206 103 155 233 350 175 263 395 593 890 445 668 334 167 251 377 566 283 425
638 319 479 719 1079 1619 2429 3644 1822 911 1367 2051 3077 4616 2308 1154 577
866 433 650 325 488 244 122 61 92 46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 92
46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 93
140 70 35 53 80 40 20 10 5 8 4 2 1
start x: 94
47 71 107 161 242 121 182 91 137 206 103 155 233 350 175 263 395 593 890 445 668
334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911 1367
2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53 80 40
20 10 5 8 4 2 1
start x: 95
143 215 323 485 728 364 182 91 137 206 103 155 233 350 175 263 395 593 890 445
668 334 167 251 377 566 283 425 638 319 479 719 1079 1619 2429 3644 1822 911
1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122 61 92 46 23 35 53
80 40 20 10 5 8 4 2 1
start x: 96
48 24 12 6 3 5 8 4 2 1
start x: 97
146 73 110 55 83 125 188 94 47 71 107 161 242 121 182 91 137 206 103 155 233 350
175 263 395 593 890 445 668 334 167 251 377 566 283 425 638 319 479 719 1079 1619
2429 3644 1822 911 1367 2051 3077 4616 2308 1154 577 866 433 650 325 488 244 122
61 92 46 23 35 53 80 40 20 10 5 8 4 2 1
start x: 98
49 74 37 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1
start x: 99
149 224 112 56 28 14 7 11 17 26 13 20 10 5 8 4 2 1

#509447

By glinert - 14 Nov 2004 06:44

- 14 Nov 2004 06:44 #509447

Even numbers always go to 1 and odd numbers if 3n+1 will go even to eventually and go down to one.

+#509467

By arcis - 14 Nov 2004 07:34

- 14 Nov 2004 07:34 #509467

"Even numbers always go to 1"

Even numbers will decrease the result of the sequence (go to one) until the result of dividing by two is also even. But there are enough cases, where the result of dividing by 2 is odd.

6, 10, 14, 18, 22, 26 ....

In this case the result of the sequence in the next step will grow.

#511305

By ThereBeDragons - 16 Nov 2004 06:40

- 16 Nov 2004 06:40 #511305

This is Goldbach's conjecture. Mathematicians (and probably loads of high school students) have spend countless hours trying to prove this. I think there's a 1 million USD reward for proving it.

Dave, Potemkin, Ombrageux: the three pillars of Western civilization. - Donald

#511351

By Iain - 16 Nov 2004 09:50

- 16 Nov 2004 09:50 #511351

I haven't got my scientific calculator with me right now, but I should be able to sort that one out this afternoon. Promise I'll buy everyone a beer with the million dollars

#511458

By bradley - 16 Nov 2004 14:36

- 16 Nov 2004 14:36 #511458

use your mobile phone, it should have a calc on that

Ixabert wrote: Why dost thou roll thine eyes at me?

#511464

By Todd D. - 16 Nov 2004 14:59

- 16 Nov 2004 14:59 #511464

Meh, it's just a math trick.

Even the numbers that when divided by two, by virtue of multiplying by 3 and adding one, will get to an integer that will half into an even, you're it's just a matter or adding your way until you get to a power of 2. Eventually you are just slowly working your way back to 2, which of course divided by 2 will be 1.

"Never put passions ahead of principles. Even if you win, you lose."

+#511580

By arcis - 16 Nov 2004 18:05

- 16 Nov 2004 18:05 #511580

Here is a really looooooooooooooooong and diiiiiirty number, but it returns to 1.

Start with:
69696969696969696969696969696969696969696969696969696969696969696969

104545454545454545454545454545454545454545454545454545454545454545454
52272727272727272727272727272727272727272727272727272727272727272727
78409090909090909090909090909090909090909090909090909090909090909091
117613636363636363636363636363636363636363636363636363636363636363637
176420454545454545454545454545454545454545454545454545454545454545456
88210227272727272727272727272727272727272727272727272727272727272728
44105113636363636363636363636363636363636363636363636363636363636364
22052556818181818181818181818181818181818181818181818181818181818182
11026278409090909090909090909090909090909090909090909090909090909091
16539417613636363636363636363636363636363636363636363636363636363637
24809126420454545454545454545454545454545454545454545454545454545456
12404563210227272727272727272727272727272727272727272727272727272728
6202281605113636363636363636363636363636363636363636363636363636364
3101140802556818181818181818181818181818181818181818181818181818182
1550570401278409090909090909090909090909090909090909090909090909091
2325855601917613636363636363636363636363636363636363636363636363637
3488783402876420454545454545454545454545454545454545454545454545456
1744391701438210227272727272727272727272727272727272727272727272728
872195850719105113636363636363636363636363636363636363636363636364
436097925359552556818181818181818181818181818181818181818181818182
218048962679776278409090909090909090909090909090909090909090909091
327073444019664417613636363636363636363636363636363636363636363637
490610166029496626420454545454545454545454545454545454545454545456
245305083014748313210227272727272727272727272727272727272727272728
122652541507374156605113636363636363636363636363636363636363636364
61326270753687078302556818181818181818181818181818181818181818182
30663135376843539151278409090909090909090909090909090909090909091
45994703065265308726917613636363636363636363636363636363636363637
68992054597897963090376420454545454545454545454545454545454545456
34496027298948981545188210227272727272727272727272727272727272728
17248013649474490772594105113636363636363636363636363636363636364
8624006824737245386297052556818181818181818181818181818181818182
4312003412368622693148526278409090909090909090909090909090909091
6468005118552934039722789417613636363636363636363636363636363637
9702007677829401059584184126420454545454545454545454545454545456
4851003838914700529792092063210227272727272727272727272727272728
2425501919457350264896046031605113636363636363636363636363636364
1212750959728675132448023015802556818181818181818181818181818182
606375479864337566224011507901278409090909090909090909090909091
909563219796506349336017261851917613636363636363636363636363637
1364344829694759524004025892777876420454545454545454545454545456
682172414847379762002012946388938210227272727272727272727272728
341086207423689881001006473194469105113636363636363636363636364
170543103711844940500503236597234552556818181818181818181818182
85271551855922470250251618298617276278409090909090909090909091
127907327783883705375377427447925914417613636363636363636363637
191860991675825558063066141171888871626420454545454545454545456
95930495837912779031533070585944435813210227272727272727272728
47965247918956389515766535292972217906605113636363636363636364
23982623959478194757883267646486108953302556818181818181818182
11991311979739097378941633823243054476651278409090909090909091
17986967969608646068412450734864581714976917613636363636363637
26980451954412969102618676102296872572465376420454545454545456
13490225977206484551309338051148436286232688210227272727272728
6745112988603242275654669025574218143116344105113636363636364
3372556494301621137827334512787109071558172052556818181818182
1686278247150810568913667256393554535779086026278409090909091
2529417370726215853370500884590331803668629039417613636363637
3794126056089323780055751326885497705502943559126420454545456
1897063028044661890027875663442748852751471779563210227272728
948531514022330945013937831721374426375735889781605113636364
474265757011165472506968915860687213187867944890802556818182
237132878505582736253484457930343606593933972445401278409091
355699317758374104380226686895515409890900958668101917613637
533548976637561156570340030343273114836351438002152876420456
266774488318780578285170015171636557418175719001076438210228
133387244159390289142585007585818278709087859500538219105114
66693622079695144571292503792909139354543929750269109552557
100040433119542716856938755689363709031815894625403664328836
50020216559771358428469377844681854515907947312701832164418
25010108279885679214234688922340927257953973656350916082209
37515162419828518821352033383511390886930960484526374123314
18757581209914259410676016691755695443465480242263187061657
28136371814871389116014025037633543165198220363394780592486
14068185907435694558007012518816771582599110181697390296243
21102278861153541837010518778225157373898665272546085444365
31653418291730312755515778167337736060847997908819128166548
15826709145865156377757889083668868030423998954409564083274
7913354572932578188878944541834434015211999477204782041637
11870031859398867283318416812751651022817999215807173062456
5935015929699433641659208406375825511408999607903586531228
2967507964849716820829604203187912755704499803951793265614
1483753982424858410414802101593956377852249901975896632807
2225630973637287615622203152390934566778374852963844949211
3338446460455931423433304728586401850167562279445767423817
5007669690683897135149957092879602775251343419168651135726
2503834845341948567574978546439801387625671709584325567863
3755752268012922851362467819659702081438507564376488351795
5633628402019384277043701729489553122157761346564732527693
8450442603029076415565552594234329683236642019847098791540
4225221301514538207782776297117164841618321009923549395770
2112610650757269103891388148558582420809160504961774697885
3168915976135903655837082222837873631213740757442662046828
1584457988067951827918541111418936815606870378721331023414
792228994033975913959270555709468407803435189360665511707
1188343491050963870938905833564202611705152784040998267561
1782515236576445806408358750346303917557729176061497401342
891257618288222903204179375173151958778864588030748700671
1336886427432334354806269062759727938168296882046123051007
2005329641148501532209403594139591907252445323069184576511
3007994461722752298314105391209387860878667984603776864767
4511991692584128447471158086814081791318001976905665297151
6767987538876192671206737130221122686977002965358497945727
10151981308314289006810105695331684030465504448037746918591
15227971962471433510215158542997526045698256672056620377887
22841957943707150265322737814496289068547385008084930566831
34262936915560725397984106721744433602821077512127395850247
51394405373341088096976160082616650404231616268191093775371
77091608060011632145464240123924975606347424402286640663057
115637412090017448218196360185887463409521136603429960994586
57818706045008724109098180092943731704760568301714980497293
86728059067513086163647270139415597557140852452572470745940
43364029533756543081823635069707798778570426226286235372970
21682014766878271540911817534853899389285213113143117686485
32523022150317407311367726302280849083927819669714676529728
16261511075158703655683863151140424541963909834857338264864
8130755537579351827841931575570212270981954917428669132432
4065377768789675913920965787785106135490977458714334566216
2032688884394837956960482893892553067745488729357167283108
1016344442197418978480241446946276533872744364678583641554
508172221098709489240120723473138266936372182339291820777
762258331648064233860181085209707400404558273508937731166
381129165824032116930090542604853700202279136754468865583
571693748736048175395135813907280550303418705131703298375
857540623104072263092703720860920825455128057697554947563
1286310934656108394639055581291381238182692086546332421345
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1

The longest post I ever made.

#511641

By Iain - 16 Nov 2004 19:20

- 16 Nov 2004 19:20 #511641

Meh, it's just a math trick.

Even the numbers that when divided by two, by virtue of multiplying by 3 and adding one, will get to an integer that will half into an even, you're it's just a matter or adding your way until you get to a power of 2. Eventually you are just slowly working your way back to 2, which of course divided by 2 will be 1.

But can you prove it will always work?

(Just finishing of some final calculations... the million dollars is nearly mine).

#511889

By yellow_turtle - 16 Nov 2004 23:41

- 16 Nov 2004 23:41 #511889

Arcis i really, really, really, really hope that you didnt do that problem yourself. Anyways back to all this math stuff. It's too much to cram into my mind, who do you think i am, Einstein? It all ends up back to one. That's all i got out of it. Gosh now i have a headache. :knife:

freedom is not worth having if it does not include the freedom to make mistakes.

#511932

By Seán Himmelb(L)au - 17 Nov 2004 00:12

- 17 Nov 2004 00:12 #511932

I don't understand how any sequence can converge to 1 exclusively, because while some can be converging:

2/3, 3/5, 4/7, 5/9, 6/11,...

Has the formula (u^n = n+1) over (u^n = 2n+1) or cancelled down to n+1 over 2n+1, but this converges to 1/2, whereas:

4, 6, 9, 13.5, 20.25,...

Of course, you just keep multiplying by 1.5:

u^1= 4 x 1.5^0
u^2= 4 x 1.5^1
u^3= 4 x 1.5^2
u^4= 4 x 1.5^3
u^5= 4 x 1.5^4

etc. etc., so the number just diverges and multiplies into infinity or whatever... but never to 1.

So what do you mean?

I use the words you taught me. If they don't mean anything any more, teach me others. Or let me be silent. - Samuel Beckett

#512142

By glinert - 17 Nov 2004 05:08

- 17 Nov 2004 05:08 #512142

Arcis must love Window Eksel!

+#512182

By arcis - 17 Nov 2004 06:53

- 17 Nov 2004 06:53 #512182

Exel? Shame on you!
No, I am just fooling around with Java.