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

0,3

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.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A092411 A272744 A273536 * A273117 A273689 A143274

Adjacent sequences: A134503 A134504 A134505 * A134507 A134508 A134509

Keyword

nonn,nice

Author

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

Extensions

More terms from Stefan Steinerberger, Feb 06 2008

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

Status

approved