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A343860 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.

%I A343860 #63 Jun 19 2021 20:23:49
%S A343860 8,9,10,18,15,24,45,35,90,90,210,264,117,90,585,136,435,522,1305,1935,
%T A343860 306,235,3978,3608,4690,2415,1416,801,615,792,27234,1610,6090,50184,
%U A343860 44290,3042,44109,8730,22698,41615,2097,1610,107535,186633,46104,40410,19485
%N A343860 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.
%C A343860 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).
%H A343860 Soumyadeep Dhar, <a href="/A343860/b343860.txt">Table of n, a(n) for n = 1..112</a>
%e A343860 t^2 - (3+15)*t + 3*15 = 0 has roots p=3 and q=15, and
%e A343860 t^2 - (9+36)*t + 9*36 = 0 has roots u=9 and v=36, and
%e A343860 3*15 = 9+36 and (3+15)^2 = 9*36, so k = 3+15 = 18 is a term of this sequence.
%e A343860 --
%e A343860 The first 10 values of k listed in A057369 and their corresponding values of w, x, y, z, and y+z are as follows:
%e A343860 .
%e A343860    n    k   w    x  y   z  y+z = a(n)
%e A343860   --  ---  --  --- --  --  ----------
%e A343860    1   16   8    8  4   4        8
%e A343860    2   18   9    9  3   6        9
%e A343860    3   25   5   20  5   5       10
%e A343860    4   45   9   36  3  15       18
%e A343860    5   50   5   45  5  10       15
%e A343860    6   80   8   72  4  20       24
%e A343860    7  234   9  225  6  39       45
%e A343860    8  250   5  245 10  25       35
%e A343860    9  261  36  225  3  87       90
%e A343860   10  425  20  405  5  85       90
%o A343860 (PARI) forstep(k=1, 1000, 1, fordiv(k, y, if(issquare(k^2 - 4*(y+k/y)^2), print1(y+k/y, ", "); break)));
%Y A343860 Cf. A057369, A057442.
%K A343860 nonn
%O A343860 1,1
%A A343860 _Soumyadeep Dhar_, May 01 2021