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A134506
Number of 2 X 2 singular integer matrices with elements from {1,...,n}.
11
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
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