A135412 Integers that equal three times the Heronian mean of two positive integers.

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

Offset: 1

Author

Pahikkala Jussi, Feb 17 2008

Keywords

• easy
• nonn

Comments

The Heronian mean of two nonnegative real numbers x and y is (x + y + sqrt(xy))/3. Therefore any number n is the Heronian mean of x = 3n and y = 0 (and also of x = n and y = n).

In particular, the sequence contains all numbers n = 3k which equal three times the Heronian mean of k and itself. If the two integers are required to be distinct then most multiples of 3 are no longer in the sequence: see A050931 for the sequence of integers that equal the Heronian mean of two distinct positive integers. Writing x = r^2*s where s is squarefree, the square root is an integer iff y = k^2*s for some integer k, and thus n = s*(r^2 + k^2 + rk). Therefore this sequence consists of the numbers listed in A024614 and their multiples by squarefree s. - M. F. Hasler, Aug 17 2016

Examples

			35 is in the sequence since 5 + 20 + sqrt(5*20) = 35.
		

Links

• Planet Math, Heronian mean is between geometric and arithmetic mean
• Eric W. Weisstein, Heronian Mean. From MathWorld--A Wolfram Web Resource.
• Wikipedia, Heronian mean

Crossrefs

Cf. A024614, A050931, A004612.

Extensions

Edited and definition corrected, following a remark by Robert Israel, by M. F. Hasler, Aug 17 2016