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Variation on Fermat's Diophantine m-tuple: 1 + the LCM of any two distinct terms is a square.
0

%I #16 Mar 14 2020 17:14:14

%S 0,1,3,8,15,24,120,168,840,1680,5040,201600,256032000

%N Variation on Fermat's Diophantine m-tuple: 1 + the LCM of any two distinct terms is a square.

%C a(1) = 0; for n>1, a(n) = smallest integer > a(n-1) such that lcm(a(n),a(i))+1 is square for all 1 <= i <= n-1.

%e After a(1)=0, a(2)=1, a(3)=3, we want m, the smallest number > 3 such that lcm(0,m)+1, lcm(2,m)+1 and lcm(3,m)+1 are squares: this is m = 8 = a(4).

%t a = {0}; Do[AppendTo[a, SelectFirst[Range[Max@ a + 1, 3*10^5], Function[k, Times @@ Boole@ Map[IntegerQ@ Sqrt[LCM[a[[#]], k] + 1] &, Range[n - 1]] == 1]]], {n, 2, 12}]; a (* _Michael De Vlieger_, Jul 05 2016, Version 10 *)

%o (Sage)

%o seq = [0]

%o prev_element = 0

%o max_n = 13

%o for n in range(2, max_n+1):

%o next_element = prev_element + 1

%o while True:

%o all_match = True

%o for element in seq:

%o x = lcm( element, next_element ) + 1

%o if not is_square(x):

%o all_match = False

%o break

%o if all_match:

%o seq.append( next_element )

%o print(seq)

%o break

%o next_element += 1

%o prev_element = next_element

%o print(seq)

%Y Cf. A030063.

%K nonn

%O 1,3

%A _Robert C. Lyons_, Jul 05 2016