a number challenge

[MattcAnderson]

Hi all,

Here is a mathematics problem.
For which positive integers n, is there a sum of n positive integers that is a perfect square?

Source : Math horizons, September 2016, p. 31.

Some are aware that the sum of integers from 1 to n can be written as
s=n*(n+1)/2.
Also, such numbers as 1,3,6,10, ... are known as triangular numbers.
Think of the sport bowling. There are 10 bowling pins and the pins are arranged in a triangle.
Some Maple code reveals that the first few n that satisfy the above criterion are 1,8,288,1681.
Can anyone find a general form?
I did not find this sequence in the OEIS.org.

[retina]

How do you define "perfect square"? Must it be integers only? Or can it also be fractions and complex numbers etc.?

[R. Gerbicz]

Originally Posted by MattcAnderson

Hi all,

Here is a mathematics problem.
For which positive integers n, is there a sum of n positive integers that is a perfect square?

Source : Math horizons, September 2016, p. 31.

Some are aware that the sum of integers from 1 to n can be written as
s=n*(n+1)/2.
Also, such numbers as 1,3,6,10, ... are known as triangular numbers.
Think of the sport bowling. There are 10 bowling pins and the pins are arranged in a triangle.
Some Maple code reveals that the first few n that satisfy the above criterion are 1,8,288,1681.
Can anyone find a general form?
I did not find this sequence in the OEIS.org.

The original problem:
"Call a positive integer n good if the sum of n consecutive integers could be a perfect square, and bad otherwise. For example, 3 is good because 2+3+4=9=3^2. In Square sums, you were asked to find all bad numbers."

It is a quite different problem from the above, and has got a better wording. The problem is very well known.

[MattcAnderson]

Hi all,

@retina I should have posted that we want
to assume that n is an integer.

I did not want to consider fractions, irrationals, and other real numbers.

Further, I want to restrict this puzzle to the real numbers.

Complex numbers are out


Also, this problem is well known by those that well know it.
I copied it from a local University "POW" Problem Of the Week.
Luckily, I am still on their email distribution list.

Regards,
Matt

[MattcAnderson]

Hi mersenneforum

To be clear, perfect square numbers are
numbers like
0, 1, 4, 9, ...

I guess that was a definition by example

Regards
Matthew

[science_man_88]

Originally Posted by MattcAnderson

Hi all,

Here is a mathematics problem.
For which positive integers n, is there a sum of n positive integers that is a perfect square?

Source : Math horizons, September 2016, p. 31.

Some are aware that the sum of integers from 1 to n can be written as
s=n*(n+1)/2.
Also, such numbers as 1,3,6,10, ... are known as triangular numbers.
Think of the sport bowling. There are 10 bowling pins and the pins are arranged in a triangle.
Some Maple code reveals that the first few n that satisfy the above criterion are 1,8,288,1681.
Can anyone find a general form?
I did not find this sequence in the OEIS.org.

my guess basically most of them because x^2 is the sum of x numbers that average to x, so 1^2 = 1, 2^2 = 1+3, 3^2 = 2+3+4,4^2 =3+4+4+5. edit: the squares are known to be the sum of the first x odd integers as well. 1^2=1;2^2=1+3;3^2 = 1+3+5; etc.

[CRGreathouse]

Originally Posted by MattcAnderson

Some Maple code reveals that the first few n that satisfy the above criterion are 1,8,288,1681.
Can anyone find a general form?
I did not find this sequence in the OEIS.org.

I'll follow Math Horizons and call a number n "good" if there is a sum of n consecutive integers which is square.

1 is good because 1 is a square. 2 is good because 4+5 = 3^2. 3 is good because 2 + 3 + 4 = 3^2. 4 is bad because n + n+1 + n+2 + n+3 = 4n + 6 is never a square. 5 is good because 3 + 4 + 5 + 6 + 7 = 5^2. So I get a very different list from you: 4, 12, 16, 20, 28, 36, 44, 48, 52, 60, 64, 68, 76, 80, 84, 92, 100, ... which is A108269 in the OEIS.

[xilman]

Originally Posted by CRGreathouse

I'll follow Math Horizons and call a number n "good" if there is a sum of n consecutive integers which is square.

I believe that the OP meant:

Find solutions (m,n) in integers to the Diophantine equation m^2 = n(n+1)/2.

He further asserts that the the sequence of values for m is not in the OEIS.

[science_man_88]

Originally Posted by xilman

I believe that the OP meant:

Find solutions (m,n) in integers to the Diophantine equation m^2 = n(n+1)/2.

He further asserts that the the sequence of values for m is not in the OEIS.

a quick search with PARI shows those values he listed are a incomplete list of the n values actually. edit: with a more complete list of values you get https://oeis.org/A001108

[CRGreathouse]

Originally Posted by science_man_88

a quick search with PARI shows those values he listed are a incomplete list of the n values actually. edit: with a more complete list of values you get https://oeis.org/A001108

Right, or https://oeis.org/A001109 in the opposite direction.

[MattcAnderson]

Hi Mersenneforum,
Thank you for your replies. C.R.Greathouse, you seem to have figured it out. Good show.
Regards,
Matthew