A237620 Semiperimeters s of primitive Pythagorean triples (a, b, c) where a, b, c and s are not squarefree.

513, 1960, 2254, 2684, 3225, 3276, 3843, 3969, 4260, 4482, 4950, 5022, 5148, 5247, 5994, 6903, 7029, 7502, 7956, 8658, 10164, 10527, 12201, 12463, 12750, 12936, 13189, 13552, 13728, 13923, 15575, 15717, 16023, 16244, 16611, 16768

Offset

1,1

Comments

No primitive Pythagorean triangle (PPT) can have all its integer sides squarefree since at least one side must be divisible by 4. However it is possible to find PPT's where none of the integer sides and the semiperimeter are squarefree. a(n) is the ordered occurrences of such semiperimeters.

Links

Table of n, a(n) for n=1..36.

Example

a(5)=3225 as the PPT (825, 2752, 2873) has a semiperimeter of 3225, no member of (825, 2752, 2873, 3225) is squarefree and it is the 5th occurrence of such a semiperimeter.

Mathematica

getpairs[k_] := Reverse[Select[IntegerPartitions[k, {2}], GCD[#[[1]], #[[2]]]==1 &]]; getlist[j_] := (newlist=getpairs[j]; Table[{newlist[[m]][[1]]^2-newlist[[m]][[2]]^2, 2newlist[[m]][[1]]*newlist[[m]][[2]], newlist[[m]][[1]]^2+newlist[[m]][[2]]^2, newlist[[m]][[1]]*j}, {m, 1, Length[newlist]}]); p=100; quads={}; Do[AppendTo[quads, getlist[2n+1]], {n, 1, p}]; sqquads=Select[Flatten[quads, 1], Union@Table[SquareFreeQ@#[[r]], {r, 1, 4}]=={False} &]; lst=Table[sqquads[[k]][[4]], {k, 1, Length@sqquads}]; Select[Sort@lst, #<2 p^2 &]

Crossrefs

Sequence in context: A182108 A066697 A076338 * A111344 A230188 A223651

Adjacent sequences: A237617 A237618 A237619 * A237621 A237622 A237623

Keyword

nonn

Author

Frank M Jackson, Feb 10 2014

Status

approved