6) sum of the first 45 odd numbers: 1+3+5+...+89
7) is a Harshad number: https://en.m.wikipedia.org/wiki/Harshad_number
3^2 is the sum of the first three odd numbers. 4^2 is the sum of the first four odd numbers. 5^2 is the sum of the first five odd numbers.
Edit: sorry, don't mean to be a pill.
Actually, property 5) trivially implies 1) but also 2), as `(1+2+...+n)² = n²(n+1)²/4` and either n or n+1 must be divisible by 2 hence one of the squares divisible by 4 hence it is a product of squares. But also property 4) as `(1+2+...+n)² = 1³+2³+...+n³` (easy to show by induction).
There is no such thing as a Harshad number, there is a _Harshad number in a given base_. All integers between zero and n are n-harshad numbers.
Which is a pity, because apparenty it means the `joy-giver`. I think human kind could use a joy giver year
