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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

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

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

Cf. A000290 (squares).

Sequence in context: A043352 A223183 A171995 * A262798 A250431 A046755

Adjacent sequences:  A307478 A307479 A307480 * A307482 A307483 A307484

KEYWORD

nonn

AUTHOR

Glen Gilchrist, Apr 10 2019

STATUS

approved

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Last modified September 16 07:19 EDT 2021. Contains 347469 sequences. (Running on oeis4.)