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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. 1

%I

%S 0,1,2,5,17,37,101,197,257,401,577,677,901,1297,1601,2917,3137,4357,

%T 5477,7057,8101,8837,10001,12101,13457,14401,15377,15877,16901,17957,

%U 20737,21317,22501,24337,25601,28901,30977,32401,33857,41617,42437,44101,50177,52901

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

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

%H Chai Wah Wu, <a href="/A276460/b276460.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 (Python)

%o from __future__ import division

%o from sympy import divisors

%o from gmpy2 import is_square

%o A276460_list = [0]

%o for m in range(10**3):

%o k = m**2+1

%o for d in divisors(k):

%o if d > m:

%o A276460_list.append(k)

%o break

%o if not is_square(k//d - d):

%o break # _Chai Wah Wu_, Sep 04 2016

%Y Cf. A002496, A134406.

%K nonn

%O 1,3

%A _Michel Lagneau_, Sep 03 2016

%E Terms 0, 1 added by _Chai Wah Wu_, Sep 04 2016

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