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

(In reply to answer by Praneeth)

Verified gives the same answers:

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

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, p * (p + 1) * (3 * p * p - 3 * p + 2) / 2
NEXT

Posted by Charlie on 2008-05-23 00:10:42