# [seqfan] A135412

israel at math.ubc.ca israel at math.ubc.ca
Wed Aug 17 19:08:58 CEST 2016

```The Name is: Integers that are Heronian means of two distinct nonnegative
integers. Data: 3, 6, 7, 9, 12, 13, 14, 15, 18, 19, 21, 24, 26, 27, 28, 30,
31, 33, 35, 36, 37, 38, 39, 42, 43, 45, 48, 49, 51, 52, 54, 56, 57, 60, 61,
62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 81, 84, 86, 87,
90, 91, 93, 95, 96, 97, 98, 99, 102, 103, 104, 105, 108, 109, 111

The Comment is: If "nonnegative" is changed to "positive", most numbers
divisible by 3 drop out of the sequence.

But if 0's are allowed, why isn't every positive integer in this sequence?
n is the Heronian mean of 0 and 3*n.

Meanwhile, the given Data seems to match a different sequence: the set of
positive integers n such that for some integer c, x^3 + n x^2 + c has three
distinct integer roots. Equivalently, n is in the sequence iff there exist
distinct integers r1, r2, r3 such that r1 + r2 + r3 = n and r1 r2 + r1 r3 +
r2 r3 = 0.

Cheers,
Robert
```