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A343860
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For the numbers k that can be expressed as k = w+x = y*z with w*x = (y+z)^2 where w, x, y, and z are all positive integers, this sequence gives the corresponding values of y+z.
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1
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8, 9, 10, 18, 15, 24, 45, 35, 90, 90, 210, 264, 117, 90, 585, 136, 435, 522, 1305, 1935, 306, 235, 3978, 3608, 4690, 2415, 1416, 801, 615, 792, 27234, 1610, 6090, 50184, 44290, 3042, 44109, 8730, 22698, 41615, 2097, 1610, 107535, 186633, 46104, 40410, 19485
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OFFSET
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1,1
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COMMENTS
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A057369 lists numbers m such that two quadratic equations of the form t^2-k*t+m = 0 and t^2-m*t+k^2 = 0 have positive integer roots, where k is the coefficient of t and m is the constant in first equation, which has roots p and q (i.e., k, m, p, q are all positive integer, k=p+q and m=p*q). Also m is the coefficient of t and k^2 is the constant in second equation, which has roots u and v (i.e., k, m, u, v are all positive integer, m=u+v and k^2=u*v). Sequence [a(n)] represents corresponding values of k=p+q for A057369(m).
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LINKS
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EXAMPLE
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t^2 - (3+15)*t + 3*15 = 0 has roots p=3 and q=15, and
t^2 - (9+36)*t + 9*36 = 0 has roots u=9 and v=36, and
3*15 = 9+36 and (3+15)^2 = 9*36, so k = 3+15 = 18 is a term of this sequence.
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The first 10 values of k listed in A057369 and their corresponding values of w, x, y, z, and y+z are as follows:
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n k w x y z y+z = a(n)
-- --- -- --- -- -- ----------
1 16 8 8 4 4 8
2 18 9 9 3 6 9
3 25 5 20 5 5 10
4 45 9 36 3 15 18
5 50 5 45 5 10 15
6 80 8 72 4 20 24
7 234 9 225 6 39 45
8 250 5 245 10 25 35
9 261 36 225 3 87 90
10 425 20 405 5 85 90
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PROG
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(PARI) forstep(k=1, 1000, 1, fordiv(k, y, if(issquare(k^2 - 4*(y+k/y)^2), print1(y+k/y, ", "); break)));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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