A134506 Number of 2 X 2 singular integer matrices with elements from {1,...,n}.

0, 1, 6, 15, 32, 49, 86, 111, 160, 209, 278, 319, 432, 481, 582, 703, 832, 897, 1078, 1151, 1360, 1537, 1702, 1791, 2096, 2257, 2454, 2671, 2976, 3089, 3510, 3631, 3952, 4241, 4502, 4831, 5360, 5505, 5798, 6143, 6704, 6865, 7478, 7647, 8144, 8721, 9078, 9263

Offset: 0

Author

Graziano Aglietti (mg5055(AT)mclink.it), Jan 20 2008, Feb 04 2008

Keywords

• nonn
• nice

Comments

a(2k) is even. a(4k+i) = i (mod 4), for i = 0, 1, 2, 3. - Aldo González Lorenzo, Oct 14 2011

Links

• Charles R Greathouse IV and Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms for n = 1..1000 from Charles R Greathouse IV)
• Sanying Shi, On the equation n1n2 = n3n4 and mean value of character sums, Journal of Number Theory, Volume 128, Issue 2, February 2008, Pages 313-321.

Crossrefs

Cf. A059306 (similar but with elements from {0, ..., n}).

Programs

Mathematica

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 *)

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

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

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

Formula

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

a(n) = A059306(n) - (2n+1)^2. - Chai Wah Wu, Nov 28 2016

Extensions

More terms from Stefan Steinerberger, Feb 06 2008

a(0) added by Chai Wah Wu, Nov 28 2016