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

%I A135412 #34 Feb 16 2025 08:33:07
%S A135412 3,6,7,9,12,13,14,15,18,19,21,24,26,27,28,30,31,33,35,36,37,38,39,42,
%T A135412 43,45,48,49,51,52,54,56,57,60,61,62,63,65,66,67,69,70,72,73,74,75,76,
%U A135412 77,78,79,81,84,86,87,90,91,93,95,96,97,98,99,102,103,104,105,108,109,111
%N A135412 Integers that equal three times the Heronian mean of two positive integers.
%C A135412 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).
%C A135412 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
%H A135412 Planet Math, <a href="http://planetmath.org/heronianmeanisbetweengeometricandarithmeticmean">Heronian mean is between geometric and arithmetic mean</a>
%H A135412 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/HeronianMean.html">Heronian Mean.</a> From MathWorld--A Wolfram Web Resource.
%H A135412 Wikipedia, <a href="http://en.wikipedia.org/wiki/Heronian_mean">Heronian mean</a>
%e A135412 35 is in the sequence since 5 + 20 + sqrt(5*20) = 35.
%Y A135412 Cf. A024614, A050931, A004612.
%K A135412 easy,nonn
%O A135412 1,1
%A A135412 _Pahikkala Jussi_, Feb 17 2008
%E A135412 Edited and definition corrected, following a remark by _Robert Israel_, by _M. F. Hasler_, Aug 17 2016