A276460 Numbers k such that for any positive integers a < b, if a * b = k then b - a is a square.

0, 1, 2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 901, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 10001, 12101, 13457, 14401, 15377, 15877, 16901, 17957, 20737, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857, 41617, 42437, 44101, 50177, 52901

Offset: 1

Author

Michel Lagneau, Sep 03 2016

Keywords

• nonn

Comments

A majority of numbers are primes of form m^2+1 (A002496), and it appears that the composite numbers of the form m^2+1: 901, 10001, 20737, 75077, 234257, 266257, 276677, 571537,... are semiprimes.

For n >1, a(n)==1,5 mod 12 and a(n)==1,5 mod 16.

Examples

			901 is in the sequence because 901 = 1*901 = 17*53 => 901-1 = 30^2 and 53-17 = 6^2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002496, A134406.

Programs

Mathematica

t={};Do[ds=Divisors[n];If[EvenQ[Length[ds]],ok=True;k=1;While[k<=Length[ds]/2&&(ok=IntegerQ[Sqrt[Abs[ds[[k]]-ds[[-k]]]]]),k++];If[ok,AppendTo[t,n]]],{n,2,10^5}];t

Python

from _future_ import division
from sympy import divisors
from gmpy2 import is_square
A276460_list = [0]
for m in range(10**3):
    k = m**2+1
    for d in divisors(k):
        if d > m:
            A276460_list.append(k)
            break
        if not is_square(k//d - d):
            break # Chai Wah Wu, Sep 04 2016

Extensions

Terms 0, 1 added by Chai Wah Wu, Sep 04 2016