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Sum Greatest And Least Roots (Posted on 2008-05-22) Difficulty: 2 of 5
Derive a formula for evaluating the following expression in terms of p and its higher powers, given that p is a positive integer.
   Σ  ([3√m] + <3√m>) 
m=1 to p3
Note: [x] is the greatest integer ≤ x, and <x> is the least integer ≥ x

See The Solution Submitted by K Sengupta    
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Some Thoughts thoughts | Comment 1 of 3

When m is a perfect cube, the two numbers being added into the summation are equal, as the cube root of m. In between, perfect cubes, the two numbers  being added in differ by 1. The last m will in fact be a perfect cube, i.e., p^3.

The following table shows, first, the last row to stop on for a given p. The next column shows the m value or range of m values for which it is valid. The next column shows how many m values are in that range. The next column is a sum, showing the two numbers that are added for that m and the total of those values. The next column multiplies that total by the number of m values that have that total (i.e., the third column). And finally the last column shows the cumulative total thus far for valid p values:

p    m range   how many m's    pair sum     total value    cum. total
1       1           1           1+1=2            2             2
       2-7          6           1+2=3           18    
2       8           1           2+2=4            4            24
       9-26        18           2+3=5           90            
3       27          1           3+3=6            6           120

...


The alternate terms are in arithmetic progression and are easy enough to evaluate. The intermediate ones are more difficult.

The overall summations for several values of p are:

1             2
2             24
3             120
4             380
5             930
6             1932
7             3584
8             6120
9             9810
10            14960
11            21912
12            31044
13            42770
14            57540
15            75840
16            98192
17            125154
18            157320
19            195320
20            239820

calculated by

DEFDBL A-Z
CLS
FOR p = 1 TO 20
 t = 0
 FOR i = 1 TO p * p * p
   cr = INT(i ^ (1 / 3) + .5)
   IF cr * cr * cr <> i THEN cr = i ^ (1 / 3)
   t = t + INT(cr) - INT(-cr)
 NEXT
 PRINT p, t
NEXT

Note the first three values agree with the cum. totals given in the manually drawn table.

I'd imagine there'd be a polynomial for these but I haven't found it using difference tables.


  Posted by Charlie on 2008-05-22 17:23:07
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