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%I #23 Nov 29 2016 08:00:17
%S 0,1,6,15,32,49,86,111,160,209,278,319,432,481,582,703,832,897,1078,
%T 1151,1360,1537,1702,1791,2096,2257,2454,2671,2976,3089,3510,3631,
%U 3952,4241,4502,4831,5360,5505,5798,6143,6704,6865,7478,7647,8144,8721,9078,9263
%N Number of 2 X 2 singular integer matrices with elements from {1,...,n}.
%C a(2k) is even. a(4k+i) = i (mod 4), for i = 0, 1, 2, 3. - _Aldo González Lorenzo_, Oct 14 2011
%H Charles R Greathouse IV and Chai Wah Wu, <a href="/A134506/b134506.txt">Table of n, a(n) for n = 0..10000</a> (terms for n = 1..1000 from Charles R Greathouse IV)
%H Sanying Shi, <a href="http://dx.doi.org/10.1016/j.jnt.2007.10.004">On the equation n1n2 = n3n4 and mean value of character sums</a>, Journal of Number Theory, Volume 128, Issue 2, February 2008, Pages 313-321.
%F Shi proves that a(n) = kn^2 log n + cn^2 + O(n^e) where k = 12/Pi^2, e > 547/416 = 1.3149..., and c is a complicated constant given in the paper (see p. 320 and pp. 314-315). - _Charles R Greathouse IV_, Feb 03 2016
%F a(n) = A059306(n) - (2n+1)^2. - _Chai Wah Wu_, Nov 28 2016
%t a = {}; For[n = 2, n < 50, n++, s = 0; For[j = 1, j < n + 1, j++, For[c = 1, c < n + 1, c++, s = s + Length[Select[Divisors[c*j], # < n + 1 && c*j/# < n + 1 &]]]]; AppendTo[a, s]]; a (* _Stefan Steinerberger_, Feb 06 2008 *)
%o (PARI) a(n) = {my(nnb = 0); for (i=1, n, for (j=1, n, pij = i*j; for (k=1, n, for (l=1, n, if (pij == k*l, nnb++););););); nnb;} \\ _Michel Marcus_, Feb 03 2016
%o (PARI) a(n)=sum(i=1,n,sum(j=1,n,my(ij=i*j);sumdiv(ij,k, k<=n && ij/k<=n))) \\ _Charles R Greathouse IV_, Feb 03 2016
%o (PARI) a(n)=2*sum(i=2,n,sum(j=1,i-1,my(ij=i*j);sumdiv(ij,k, k<=n && ij/k<=n))) + sum(i=1,n,my(i2=i^2);sumdiv(i2,k, k<=n && i2/k<=n)) \\ _Charles R Greathouse IV_, Feb 03 2016
%Y Cf. A059306 (similar but with elements from {0, ..., n}.
%K nonn,nice
%O 0,3
%A Graziano Aglietti (mg5055(AT)mclink.it), Jan 20 2008, Feb 04 2008
%E More terms from _Stefan Steinerberger_, Feb 06 2008
%E a(0) added by _Chai Wah Wu_, Nov 28 2016
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