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.

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

Offset

1,1

Comments

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).

Links

Soumyadeep Dhar, Table of n, a(n) for n = 1..112

Example

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.
--
The first 10 values of k listed in A057369 and their corresponding values of w, x, y, z, and y+z are as follows:
.
   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

Prog

(PARI) forstep(k=1, 1000, 1, fordiv(k, y, if(issquare(k^2 - 4*(y+k/y)^2), print1(y+k/y, ", "); break)));

Crossrefs

Cf. A057369, A057442.

Sequence in context: A130881 A235399 A283628 * A308809 A272142 A270039

Adjacent sequences: A343857 A343858 A343859 * A343861 A343862 A343863

Keyword

nonn

Author

Soumyadeep Dhar, May 01 2021

Status

approved