A224875 Array A(n,k), n > 0, k=1,2,3, read by rows such that A(n,i) + A(n,j) is a square for i <> j with A(n,1) = n.

1, 8, 8, 2, 2, 2, 3, 22, 78, 4, 21, 60, 5, 20, 44, 6, 19, 30, 7, 2, 2, 8, 1, 8, 9, 40, 216, 10, 39, 186, 11, 38, 158, 12, 37, 132, 13, 36, 108, 14, 2, 2, 15, 34, 66, 16, 33, 48, 17, 8, 8, 18, 7, 18, 19, 6, 30, 20, 5, 44, 21, 4, 60, 22, 3, 78, 23, 2, 2, 24

Offset

1,2

Comments

Problem No. 40 from P. Erdős (see the first reference). The problem is "can we find for each positive integer k, k integers a1, a2, ..., ak such that the sums ai + aj are squares?” This problem is possible for k <= 4.

In this sequence we consider k = 3, a1 = n and a2 minimum.

Links

Michel Lagneau, Table of triples (n, b, c) for n = 1..1000

P. Erdős, Some unsolved problems, Publ. Inst. Hung. Acad. Sci. 6 (1961), 221-259.

P. Erdős, Quelques problèmes de théorie des nombres (in French), Monographies de l'Enseignement Mathématique, No. 6, pp. 81-135, L'Enseignement Mathématique, Université, Geneva, 1963.

Example

Array A(n,k) begins:
[1, 8, 8]
[2, 2, 2]
[3, 22, 78]
[4, 21, 60]
[5, 20, 44]
...
Row 4 = [4, 21, 60] because 4 + 21 = 5^2, 4 + 60 = 8^2 and 21 + 60 = 9^2.

Maple

with(numtheory): T:=array(1..121):k:=1:nn:=121:C:=array(1..nn):for i from 1 to nn do:C[i]:=i^2:od:for n from 1 to nn do:ii:=0:for a from 1 to nn while(ii=0) do: aa:=C[a]-n: for b from 1 to nn while(ii=0) do: bb:=C[b]-n:if aa>0 and bb>0 and sqrt(aa+bb)=floor(sqrt(aa+bb)) then ii:=1:T[k]:=n: T[k+1]:=aa: T[k+2]:=bb:k:=k+3:else fi:od:od:od:print(T):

Crossrefs

Sequence in context: A199597 A366149 A197848 * A242588 A105193 A178678

Adjacent sequences: A224872 A224873 A224874 * A224876 A224877 A224878

Keyword

nonn,tabf

Author

Michel Lagneau, Jul 23 2013

Status

approved