Number Gossip

(Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.)

Unique Properties of 1575

1575 is the smallest odd number n such that n! divided by the product of the factorials of all the proper divisors of n is not an integer

Common Properties of 1575

Abundant

The number n is abundant if the sum of all its positive divisors except itself is more than n.

They are abundant above perfection, not to mention deficiency. See perfect and deficient numbers.

...,
1566,
1568,
1572,
1575,
1578,
1580,
1584,
...

Composite

A positive integer greater than 1 that is not prime is called composite.

Composite numbers are opposite to prime numbers.

...,
1572,
1573,
1574,
1575,
1576,
1577,
1578,
...

Evil

The number n is evil if it has an even number of 1's in its binary expansion.

Guess what odious numbers are.

...,
1569,
1570,
1572,
1575,
1576,
1579,
1581,
...

Happy

One can take the sum of the squares of the digits of a number. Those numbers are happy for which iterating this operation eventually leads to 1.

...,
1557,
1572,
1574,
1575,
1578,
1581,
1582,
...

Lucky

To build the lucky number sequence, start with natural numbers. Delete every second number, leaving 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, ... . The second number remaining is 3, so delete every third number, leaving 1, 3, 7, 9, 13, 15, 19, 21, ... . The next number remaining is 7, so delete every 7th number, leaving 1, 3, 7, 9, 13, 15, 21, ... . The next number remaining is 9, so delete every ninth number, etc.

Those numbers were lucky they weren't crossed out.

...,
1545,
1563,
1567,
1575,
1579,
1585,
1587,
...

Odd

A number is odd if it is not divisible by 2.

Numbers that are not odd are even. Compare with another pair -- evil and odious numbers.

...,
1569,
1571,
1573,
1575,
1577,
1579,
1581,
...

Tanya Khovanova

Number Gossip

Unique Properties of 1575

Common Properties of 1575