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 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

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 computer solution Comment 3 of 3 |
After first getting a program to find the first solution, it was quite slow in searching larger numbers.  Having then seen Charlie's solution and Sloane's A189173, I made the algorithm more efficient and ran the loop up to 100,000.  I found the first 3 solutions (it took about 40 minutes):

n  starting number
47       22
2161    989
99359 45450

(Python):

def sumSqrs(f,n):
"""  Starting with f, consider f,f+1,f+2, ... , f+n-1
take their squares
output the sum """
ans=0
for i in range(f,f+n):
ans += i**2
return ans

lastNum =  100000
foundOne = False
guessRatio = .5

for n in range (2,lastNum):
goal = n**3
guessFirstNumber = int(n*guessRatio) + 1

sumNow = sumSqrs(guessFirstNumber, n)
looking = True
while looking:
if sumNow == goal:
print(n,guessFirstNumber)
foundOne = True
guessRatio = guessFirstNumber / n
break
elif sumNow > goal:
guessFirstNumber -= 1
sumNow = sumNow -  2*n*guessFirstNumber - n**2
elif sumNow < goal:
# went too far, didn' find one for this n
looking = False

Edited on July 17, 2020, 12:11 pm
 Posted by Larry on 2020-07-17 12:10:51

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