Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."
After row 1, row n is obtained by inserting n between every pair of consecutive numbers that sum to n.
row 1: 1 1
row 2: 1 2 1
row 3: 1 3 2 3 1
row 4: 1 4 3 2 3 4 1
row 5: 1 5 4 3 5 2 5 3 4 5 1
row 6: 1 6 5 4 3 5 2 5 3 4 5 6 1
row 7: 1 7 6 5 4 7 3 5 7 2 7 5 3 7 4 5 6 7 1
The number of numbers in each row are 2, 3, 5, 7, 11, 13, 19...
Will they always be prime?
I wouldn't think the primes would continue, as I didn't see a relationship to prime numbers. As it turns out, primes are overrepresented for some reason, but indeed, composite numbers do show up:
The first composite occurs after inserting 10's, when there are 33 numbers in the row:
1 2 1 3 = 3
1 3 2 3 1 5 = 5
1 4 3 2 3 4 1 7 = 7
1 5 4 3 5 2 5 3 4 5 1 11 = 11
1 6 5 4 3 5 2 5 3 4 5 6 1 13 = 13
1 7 6 5 4 7 3 5 7 2 7 5 3 7 4 5 6 7 1 19 = 19
1 8 7 6 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 6 7 8 1 23 = 23
1 9 8 7 6 5 9 4 7 3 8 5 7 9 2 9 7 5 8 3 7 4 9 5 6 7 8 9 1 29 = 29
1 10 9 8 7 6 5 9 4 7 10 3 8 5 7 9 2 9 7 5 8 3 10 7 4 9 5 6 7 8 9 10 1 33 = 3 * 11
1 11 10 9 8 7 6 11 5 9 4 11 7 10 3 11 8 5 7 9 11 2 11 9 7 5 8 11 3 10 7 11 4 9 5 11 6 7 8 9 10 11 1 43 = 43
Starting with the insertion of 12's, the number of numbers is listed below. Primes do get rarer, though more common than if by chance.
12 47 = 47
13 59 = 59
14 65 = 5 * 13
15 73 = 73
16 81 = 3^4
17 97 = 97
18 103 = 103
19 121 = 11^2
20 129 = 3 * 43
21 141 = 3 * 47
22 151 = 151
23 173 = 173
24 181 = 181
25 201 = 3 * 67
26 213 = 3 * 71
27 231 = 3 * 7 * 11
28 243 = 3^5
29 271 = 271
30 279 = 3^2 * 31
31 309 = 3 * 103
32 325 = 5^2 * 13
33 345 = 3 * 5 * 23
34 361 = 19^2
35 385 = 5 * 7 * 11
36 397 = 397
37 433 = 433
38 451 = 11 * 41
39 475 = 5^2 * 19
40 491 = 491
41 531 = 3^2 * 59
42 543 = 3 * 181
43 585 = 3^2 * 5 * 13
44 605 = 5 * 11^2
45 629 = 17 * 37
46 651 = 3 * 7 * 31
47 697 = 17 * 41
48 713 = 23 * 31
49 755 = 5 * 151
50 775 = 5^2 * 31
DefDbl A-Z
Dim crlf$, nbr(1, 1000000), fct(20, 1)
Private Sub Form_Load()
Text1.Text = ""
crlf$ = Chr(13) + Chr(10)
Form1.Visible = True
nbr(0, 1) = 1: nbr(0, 2) = 1
For round = 2 To 50
nbr(1, 1) = nbr(0, 1)
ptr0 = 1: ptr1 = 1
Do
ptr0 = ptr0 + 1: ptr1 = ptr1 + 1
If nbr(0, ptr0 - 1) + nbr(0, ptr0) = round Then
nbr(1, ptr1) = round
ptr1 = ptr1 + 1
End If
nbr(1, ptr1) = nbr(0, ptr0)
Loop Until nbr(0, ptr0 + 1) = 0
For i = 1 To ptr1
nbr(0, i) = nbr(1, i)
If round < 12 Or i = 2 Then Text1.Text = Text1.Text & Str(nbr(0, i))
Next
DoEvents
Text1.Text = Text1.Text & " " & Str(ptr1) & " = "
DoEvents
f = factor(ptr1)
For i = 1 To f
Text1.Text = Text1.Text & fct(i, 0)
If fct(i, 1) > 1 Then Text1.Text = Text1.Text & "^" & fct(i, 1)
If i < f Then
Text1.Text = Text1.Text & " * "
End If
Next
Text1.Text = Text1.Text & crlf
Next round
End Sub
Function factor(num)
diffCt = 0: good = 1
nm1 = Abs(num): If nm1 > 0 Then limit = Sqr(nm1) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
dv = 2: GoSub DivideIt
dv = 3: GoSub DivideIt
dv = 5: GoSub DivideIt
dv = 7
Do Until dv > limit
GoSub DivideIt: dv = dv + 4 '11
GoSub DivideIt: dv = dv + 2 '13
GoSub DivideIt: dv = dv + 4 '17
GoSub DivideIt: dv = dv + 2 '19
GoSub DivideIt: dv = dv + 4 '23
GoSub DivideIt: dv = dv + 6 '29
GoSub DivideIt: dv = dv + 2 '31
GoSub DivideIt: dv = dv + 6 '37
If INKEY$ = Chr$(27) Then s$ = Chr$(27): Exit Function
Loop
If nm1 > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = nm1: fct(diffCt, 1) = 1
factor = diffCt
Exit Function
DivideIt:
cnt = 0
Do
q = Int(nm1 / dv)
If q * dv = nm1 And nm1 > 0 Then
nm1 = q: cnt = cnt + 1: If nm1 > 0 Then limit = Sqr(nm1) Else limit = 0
If limit <> Int(limit) Then limit = Int(limit + 1)
Else
Exit Do
End If
Loop
If cnt > 0 Then
diffCt = diffCt + 1
fct(diffCt, 0) = dv
fct(diffCt, 1) = cnt
End If
Return
End Function
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Posted by Charlie
on 2017-09-19 10:54:29 |
first thought and computer vindication
