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.

24, 120, 360, 840, 960, 1680, 3024, 4224, 5040, 7920, 11880, 17160, 22800, 24024, 32760, 36480, 43680, 57120, 70224, 73440, 83520, 93024, 116280, 121800, 143640, 175560, 201600, 212520, 241080, 255024, 303600, 330624, 358800, 421200, 491400, 570024, 591360

Offset: 1

Author

Ely Golden, May 03 2024

Keywords

• nonn

Comments

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.

Examples

			120 is a term since 120 = 15*8 = (4^2 - 1)*(3^2 - 1) and 120 = 11^2 - 1.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..19120 (first 408 terms from Ely Golden, terms <= 10^17)
• David A. Corneth, PARI program

Crossrefs

Intersection of A005563 and A063066.

Programs

Mathematica

Rest[Take[With[{k2=Range[500]^2-1},Select[Union[Times@@@Subsets[k2,{2}]],IntegerQ[Sqrt[#+1]]&]],50]] (* Harvey P. Dale, Apr 20 2025 *)

PARI

isok1(k) = issquare(k+1);
isok2(k) = fordiv(k, d, if (isok1(d) && isok1(k/d), return(1)));
isok(k) = isok1(k) && isok2(k); \\ Michel Marcus, May 04 2024

Python

from math import isqrt
def is_perfect_square(n): return isqrt(abs(n))**2 == n
limit = 10**17
sequence_entries = set()
for a in range(2, isqrt(isqrt(limit))+1):
    u = a**2 - 1
    for b in range(a+1, isqrt(limit//u+1)+1):
        v = b**2 - 1
        if(is_perfect_square(u*v + 1)): sequence_entries.add(u*v)
sequence_entries = sorted(sequence_entries)
for i, j in enumerate(sequence_entries, 1):
    print(i, j)

Extensions

Definition clarified by Harvey P. Dale, Apr 20 2025