A372496 Integers of the form k^2 + 1, where k >= 1, that are the product of two other integers of the form k^2 + 1, where k >= 1.

10, 50, 170, 290, 325, 442, 962, 1850, 2210, 3250, 5330, 8282, 9802, 12322, 15130, 17425, 17690, 24650, 33490, 44522, 58082, 58565, 64010, 65026, 74530, 94250, 103685, 117650, 145162, 177242, 191845, 214370, 237170, 257050, 305810, 332930, 361202

Offset

1,1

Comments

This sequence is the sequence of possible c^2 + 1 values of all triples of positive integers (a,b,c) such that (a^2 + 1)*(b^2 + 1) = c^2 + 1.

A001541(n)^2 + 1 is always a term of this sequence for all n > 0. This is because A001541 consists of the x-values of the solutions to the Pell equation x^2 - 2y^2 = 1 which is equivalent to x^2 + 1 = 2*(y^2 + 1) = (1^2 + 1)*(y^2 + 1).

A002061(n)^2 + 1 is always a term of this sequence for all n > 1. This is because A002061 consists of the integers of the form k^2 + k + 1 and (k^2 + k + 1)^2 + 1 = (k^2 + 1)*((k+1)^2 + 1).

Links

Ely Golden, Table of n, a(n) for n = 1..10667 (terms <= 10^16)

Example

50 is a term since 50 = 7^2 + 1 = 10 * 5 = (3^2 + 1)*(2^2 + 1).

Mathematica

formQ[k_] := k >= 1 && IntegerQ@Sqrt[k - 1];
prodQ[k_] := AnyTrue[Divisors[k][[2 ;; -2]], formQ[#] && formQ[k/#]&];
okQ[k_] := formQ[k] && prodQ[k];
Select[Range[2, 10^6], okQ] (* Jean-François Alcover, May 10 2024 *)

Prog

(Python)
from math import isqrt
def is_perfect_square(n): return isqrt(abs(n))**2 == n
limit = 10**16
sequence_entries = set()
for a in range(1, isqrt(isqrt(limit))+1):
    u = a**2 + 1
    for b in range(a+1, isqrt(limit//u)+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)
(PARI) isok1(k) = issquare(k-1) && (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

Crossrefs

Intersection of A002522 and A135280.

Cf. A001541, A002061.

Sequence in context: A196507 A008531 A337732 * A051230 A008413 A006542

Adjacent sequences: A372493 A372494 A372495 * A372497 A372498 A372499

Keyword

nonn

Author

Ely Golden, May 03 2024

Status

approved