|
| |
|
|
A307481
|
|
Numbers that can be expressed as x+2y+z such that x, y, z, x+y, y+z, and x+2y+z are all positive squares.
|
|
1
|
|
|
|
625, 2500, 5625, 10000, 15625, 22500, 28561, 30625, 40000, 50625, 62500, 75625, 83521, 90000, 105625, 114244, 122500, 140625, 142129, 160000, 180625, 202500, 225625, 250000, 257049, 275625, 302500, 330625, 334084, 360000, 390625, 422500, 455625, 456976, 490000, 525625
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
Generated by iterating through all combinations of x,y,z in the range 1..5000 (squared) and completing the addition pyramid (see Example section).
If k is in the sequence then so is k*m^2 for m >= 1. - David A. Corneth, May 04 2019
If a^2 + b^2 = c^2 then x = a^4, y = (ab)^2, z = b^4 gives a term x + 2y + z = c^4. - David A. Corneth, May 07 2019
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
Each addition pyramid is built up from three numbers x, y, and z as follows:
.
x+2y+z
/ \
/ \
x+y y+z
/ \ / \
/ \ / \
x y z
.
The first two terms, a(1)=625 and a(2)=2500, are the apex values for the first two pyramids consisting entirely of squares:
.
625 2500
/ \ / \
/ \ / \
225 400 900 1600
/ \ / \ / \ / \
/ \ / \ / \ / \
81 144 256 324 576 1024
|
|
|
PROG
|
(Magma) a:=[]; for sw in [1..725] do w:=sw^2; for su in [1..Isqrt(w div 2)] do u:=su^2; v:=w-u; if IsSquare(v) then for sx in [1..Isqrt(u)] do x:=sx^2; y:=u-x; if (y gt 0) and IsSquare(y) then z:=v-y; if IsSquare(z) then a[#a+1]:=w; break su; end if; end if; end for; end if; end for; end for; a; // Jon E. Schoenfield, May 07 2019
(C++) See Links section.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
