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 Sum Greatest And Least Roots (Posted on 2008-05-22)
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 No Rating

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 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. total1       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             22             243             1204             3805             9306             19327             35848             61209             981010            1496011            2191212            3104413            4277014            5754015            7584016            9819217            12515418            15732019            19532020            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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