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

%I A276460 #11 Sep 04 2016 23:56:21
%S A276460 0,1,2,5,17,37,101,197,257,401,577,677,901,1297,1601,2917,3137,4357,
%T A276460 5477,7057,8101,8837,10001,12101,13457,14401,15377,15877,16901,17957,
%U A276460 20737,21317,22501,24337,25601,28901,30977,32401,33857,41617,42437,44101,50177,52901
%N A276460 Numbers k such that for any positive integers a < b, if a * b = k then b - a is a square.
%C A276460 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.
%C A276460 For n >1, a(n)==1,5 mod 12 and a(n)==1,5 mod 16.
%H A276460 Chai Wah Wu, <a href="/A276460/b276460.txt">Table of n, a(n) for n = 1..10000</a>
%e A276460 901 is in the sequence because 901 = 1*901 = 17*53 => 901-1 = 30^2 and 53-17 = 6^2.
%t A276460 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
%o A276460 (Python)
%o A276460 from __future__ import division
%o A276460 from sympy import divisors
%o A276460 from gmpy2 import is_square
%o A276460 A276460_list = [0]
%o A276460 for m in range(10**3):
%o A276460     k = m**2+1
%o A276460     for d in divisors(k):
%o A276460         if d > m:
%o A276460             A276460_list.append(k)
%o A276460             break
%o A276460         if not is_square(k//d - d):
%o A276460             break # _Chai Wah Wu_, Sep 04 2016
%Y A276460 Cf. A002496, A134406.
%K A276460 nonn
%O A276460 1,3
%A A276460 _Michel Lagneau_, Sep 03 2016
%E A276460 Terms 0, 1 added by _Chai Wah Wu_, Sep 04 2016