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.

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

Offset: 1

Author

Glen Gilchrist, Apr 10 2019

Keywords

• nonn

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

Examples

			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
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10575 (terms <= 3*10^10)
• David A. Corneth, Two examples in a pyramid shape
• Sean A. Irvine, Java program (github)
• Rémy Sigrist, C++ program for A307481

Crossrefs

Cf. A000290 (squares).

Programs

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.