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A135412
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Integers that equal three times the Heronian mean of two positive integers.
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3
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Eric W. Weisstein, Heronian Mean. From MathWorld--A Wolfram Web Resource.
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EXAMPLE
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35 is in the sequence since 5 + 20 + sqrt(5*20) = 35.
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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