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 3 multiples of cubes (Posted on 2017-10-24)
What are the 3 consecutive integers - such that each is divisible by a cube (other than 1^3)?

Provide the triplet of smallest numbers.

 See The Solution Submitted by Ady TZIDON No Rating

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 computer solution Comment 1 of 1
Assuming positive integers, or "smallest" in terms of absolute value:

The smallest triplet is

1375 = 5^3 * 11 divisible by 5^3 = 125
1376 = 2^5 * 43 divisible by 2^3 = 8
1377 = 3^4 * 17 divisible by 3^3 = 27

22624 = 2^5 * 7 * 101 divisible by 2^3 = 8
22625 = 5^3 * 181 divisible by 5^3 = 125
22626 = 2 * 3^3 * 419 divisible by 3^3 = 27
22627 = 11^3 * 17 divisible by 11^3 = 1331

All the triplets (and a quad) below 100,000:

1375-1377  3
1375 = 5^3 * 11
1376 = 2^5 * 43
1377 = 3^4 * 17

4374-4376  3
4374 = 2 * 3^7
4375 = 5^4 * 7
4376 = 2^3 * 547

4912-4914  3
4912 = 2^4 * 307
4913 = 17^3
4914 = 2 * 3^3 * 7 * 13

5750-5752  3
5750 = 2 * 5^3 * 23
5751 = 3^4 * 71
5752 = 2^3 * 719

6858-6860  3
6858 = 2 * 3^3 * 127
6859 = 19^3
6860 = 2^2 * 5 * 7^3

13310-13312  3
13310 = 2 * 5 * 11^3
13311 = 3^3 * 17 * 29
13312 = 2^10 * 13 (this one's divisible by 2^3, 4^3 and 8^3)

13375-13377  3
13375 = 5^3 * 107
13376 = 2^6 * 11 * 19
13377 = 3 * 7^3 * 13

16119-16121  3
16119 = 3^4 * 199
16120 = 2^3 * 5 * 13 * 31
16121 = 7^3 * 47

21248-21250  3
21248 = 2^8 * 83
21249 = 3^3 * 787
21250 = 2 * 5^4 * 17

22624-22627  4
22624 = 2^5 * 7 * 101
22625 = 5^3 * 181
22626 = 2 * 3^3 * 419
22627 = 11^3 * 17

24352-24354  3
24352 = 2^5 * 761
24353 = 7^3 * 71
24354 = 2 * 3^3 * 11 * 41

25623-25625  3
25623 = 3^3 * 13 * 73
25624 = 2^3 * 3203
25625 = 5^4 * 41

28375-28377  3
28375 = 5^3 * 227
28376 = 2^3 * 3547
28377 = 3^3 * 1051

31374-31376  3
31374 = 2 * 3^3 * 7 * 83
31375 = 5^3 * 251
31376 = 2^4 * 37 * 53

32750-32752  3
32750 = 2 * 5^3 * 131
32751 = 3^3 * 1213
32752 = 2^4 * 23 * 89

33614-33616  3
33614 = 2 * 7^5
33615 = 3^4 * 5 * 83
33616 = 2^4 * 11 * 191

40472-40474  3
40472 = 2^3 * 5059
40473 = 3^3 * 1499
40474 = 2 * 7^3 * 59

41742-41744  3
41742 = 2 * 3^3 * 773
41743 = 13^3 * 19
41744 = 2^4 * 2609

48248-48250  3
48248 = 2^3 * 37 * 163
48249 = 3^3 * 1787
48250 = 2 * 5^3 * 193

49624-49626  3
49624 = 2^3 * 6203
49625 = 5^3 * 397
49626 = 2 * 3^3 * 919

49734-49736  3
49734 = 2 * 3^4 * 307
49735 = 5 * 7^3 * 29
49736 = 2^3 * 6217

52623-52625  3
52623 = 3^3 * 1949
52624 = 2^4 * 11 * 13 * 23
52625 = 5^3 * 421

55375-55377  3
55375 = 5^3 * 443
55376 = 2^4 * 3461
55377 = 3^3 * 7 * 293

57967-57969  3
57967 = 7^3 * 13^2
57968 = 2^4 * 3623
57969 = 3^3 * 19 * 113

58374-58376  3
58374 = 2 * 3^3 * 23 * 47
58375 = 5^3 * 467
58376 = 2^3 * 7297

59750-59752  3
59750 = 2 * 5^3 * 239
59751 = 3^3 * 2213
59752 = 2^3 * 7 * 11 * 97

75248-75250  3
75248 = 2^4 * 4703
75249 = 3^4 * 929
75250 = 2 * 5^3 * 7 * 43

76624-76626  3
76624 = 2^4 * 4789
76625 = 5^3 * 613
76626 = 2 * 3^4 * 11 * 43

79623-79625  3
79623 = 3^4 * 983
79624 = 2^3 * 37 * 269
79625 = 5^3 * 7^2 * 13

82375-82377  3
82375 = 5^3 * 659
82376 = 2^3 * 7 * 1471
82377 = 3^6 * 113

85374-85376  3
85374 = 2 * 3^4 * 17 * 31
85375 = 5^3 * 683
85376 = 2^7 * 23 * 29

86750-86752  3
86750 = 2 * 5^3 * 347
86751 = 3^6 * 7 * 17
86752 = 2^5 * 2711

90207-90209  3
90207 = 3^3 * 13 * 257
90208 = 2^5 * 2819
90209 = 7^3 * 263

94471-94473  3
94471 = 13^3 * 43
94472 = 2^3 * 7^2 * 241
94473 = 3^3 * 3499

98440-98442  3
98440 = 2^3 * 5 * 23 * 107
98441 = 7^4 * 41
98442 = 2 * 3^3 * 1823

DefDbl A-Z
Dim crlf\$, fct(20, 1)

Form1.Visible = True

Text1.Text = ""
crlf = Chr\$(13) + Chr\$(10)

For n = 2 To 100000
f = factor(n)
good = 0
For i = 1 To f
If fct(i, 1) >= 3 Then good = 1: Exit For
Next
If good Then
ct = ct + 1
Else
If ct > 2 Then
Text1.Text = Text1.Text & n - ct & "-" & n - 1 & "  " & ct & crlf
For i = n - ct To n - 1
f = factor(i)
Text1.Text = Text1.Text & i & " = "
For j = 1 To f
Text1.Text = Text1.Text & fct(j, 0)
If fct(j, 1) > 1 Then Text1.Text = Text1.Text & "^" & fct(j, 1)
If j < f Then Text1.Text = Text1.Text & " * "
Next
Text1.Text = Text1.Text & crlf
Next
Text1.Text = Text1.Text & crlf
End If
ct = 0
End If
DoEvents
Next n

Text1.Text = Text1.Text & crlf & " done"

End Sub

Function factor(num)
diffCt = 0: good = 1
n = Abs(num): If n > 0 Then limit = Sqr(n) 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 n > 1 Then diffCt = diffCt + 1: fct(diffCt, 0) = n: fct(diffCt, 1) = 1
factor = diffCt
Exit Function

DivideIt:
cnt = 0
Do
q = Int(n / dv)
If q * dv = n And n > 0 Then
n = q: cnt = cnt + 1: If n > 0 Then limit = Sqr(n) 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

 Posted by Charlie on 2017-10-24 11:06:36

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