cp's OEIS Frontend

A372497 Positive integers of the form k^2 - 1 that are the product of two other distinct positive integers of the form k^2 - 1.

%I A372497 #36 Apr 20 2025 12:44:50
%S A372497 24,120,360,840,960,1680,3024,4224,5040,7920,11880,17160,22800,24024,
%T A372497 32760,36480,43680,57120,70224,73440,83520,93024,116280,121800,143640,
%U A372497 175560,201600,212520,241080,255024,303600,330624,358800,421200,491400,570024,591360
%N A372497 Positive integers of the form k^2 - 1 that are the product of two other distinct positive integers of the form k^2 - 1.
%C A372497 This sequence is the sequence of possible c^2 - 1 values of all triples (a,b,c) of integers > 1 such that (a^2 - 1)*(b^2 - 1) = c^2 - 1.
%H A372497 David A. Corneth, <a href="/A372497/b372497.txt">Table of n, a(n) for n = 1..19120</a> (first 408 terms from Ely Golden, terms <= 10^17)
%H A372497 David A. Corneth, <a href="/A372497/a372497.gp.txt">PARI program</a>
%e A372497 120 is a term since 120 = 15*8 = (4^2 - 1)*(3^2 - 1) and 120 = 11^2 - 1.
%t A372497 Rest[Take[With[{k2=Range[500]^2-1},Select[Union[Times@@@Subsets[k2,{2}]],IntegerQ[Sqrt[#+1]]&]],50]] (* _Harvey P. Dale_, Apr 20 2025 *)
%o A372497 (Python)
%o A372497 from math import isqrt
%o A372497 def is_perfect_square(n): return isqrt(abs(n))**2 == n
%o A372497 limit = 10**17
%o A372497 sequence_entries = set()
%o A372497 for a in range(2, isqrt(isqrt(limit))+1):
%o A372497     u = a**2 - 1
%o A372497     for b in range(a+1, isqrt(limit//u+1)+1):
%o A372497         v = b**2 - 1
%o A372497         if(is_perfect_square(u*v + 1)): sequence_entries.add(u*v)
%o A372497 sequence_entries = sorted(sequence_entries)
%o A372497 for i, j in enumerate(sequence_entries, 1):
%o A372497     print(i, j)
%o A372497 (PARI) isok1(k) = issquare(k+1);
%o A372497 isok2(k) = fordiv(k, d, if (isok1(d) && isok1(k/d), return(1)));
%o A372497 isok(k) = isok1(k) && isok2(k); \\ _Michel Marcus_, May 04 2024
%Y A372497 Intersection of A005563 and A063066.
%K A372497 nonn
%O A372497 1,1
%A A372497 _Ely Golden_, May 03 2024
%E A372497 Definition clarified by _Harvey P. Dale_, Apr 20 2025