 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  Squared average equals square (Posted on 2019-08-09) Find n consecutive perfect squares (n > 1)whose average is n2.

 No Solution Yet Submitted by Danish Ahmed Khan No Rating Comments: ( Back to comment list | You must be logged in to post comments.) computer solution | Comment 1 of 3
For the average to be n^2, the sum of the n squares must be n^3, so that's what we seek.

For n = 47, the sum of the squares from the 22nd square (484) through the 68th square (4624) is 103823, the 47th cube, so that, when divided by 47 to get the average, that average is 47^2.

When n is 2161, the sum of the squares from the 989th (978121) through the 3149th (9916201) square is 10091699281, the 2161st cube; when divided by 2161 to get the average that average is 2161^2.

The program output:

47 22 484 68 4624 103823
2161 989 978121 3149 9916201 10091699281

After running the program, entering 47,2161 into Sloane's OEIS finds A189173, Integers m such that m^3 is the sum of squares of m consecutive integers. The list is preceded by zero and 1 and followed by 99359, 4568353, 210044879, 9657496081, 444034774847, 20415942146881, 938689303981679, 43159292041010353, 1984388744582494559, 91238722958753739361, 4194996867358089516047, 192878617175513363998801, 8868221393206256654428799, 407745305470312292739725953, 18747415830241159209372965039. The recursive formula is a(n) = 46*a(n-1) - a(n-2) where the a values are the n values of the puzzle.

The program:

DefDbl A-Z
Dim squares(4000), sumsquares(4000), cubes(4000), crlf\$
Text1.Text = ""
crlf\$ = Chr(13) + Chr(10)
Form1.Visible = True

For i = 1 To 4000
squares(i) = i * i
sumsquares(i) = sumsquares(i - 1) + squares(i)
cubes(i) = squares(i) * i
Next

For n = 2 To 4000
For subtr = 0 To 4000 - n
cube = sumsquares(subtr + n) - sumsquares(subtr)
If cube = cubes(n) Then
Text1.Text = Text1.Text & n & Str(subtr + 1) & Str(squares(subtr + 1))
Text1.Text = Text1.Text & Str(subtr + n) & Str(squares(subtr + n)) & Str(cubes(n)) & crlf
End If
Next
Next

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

End Sub

 Posted by Charlie on 2019-08-09 12:59:34 Please log in:

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