A057369
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.
Original entry on oeis.org
16, 18, 25, 45, 50, 80, 234, 250, 261, 425, 1025, 1040, 1530, 1625, 1746, 2320, 4250, 7605, 7794, 9650, 10413, 11050, 11925, 14416, 23425, 24050, 27920, 71298, 75650, 78416, 81693, 129625, 151625, 200720, 221425, 257085, 264618, 338949, 340245, 416050, 488610
Offset: 1
a(1) = 16 = 8+8 = 4*4; 8*8 = (4+4)^2.
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is(k) = fordiv(k, y, if(issquare(k^2 - 4*(y+k/y)^2), return(1))); 0; \\ Jinyuan Wang, May 01 2021
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.
Original entry on oeis.org
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
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)
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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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forstep(k=1, 1000, 1, fordiv(k, y, if(issquare(k^2 - 4*(y+k/y)^2), print1(y+k/y, ", "); break)));
Showing 1-2 of 2 results.
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