cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A115223 Number of 3 X 3 symmetric matrices over Z(n) having determinant 0.

Original entry on oeis.org

1, 36, 261, 1408, 3225, 9396, 17101, 47104, 76545, 116100, 162261, 367488, 373321, 615636, 841725, 1638400, 1424481, 2755620, 2482597, 4540800, 4463361, 5841396, 6447981, 12294144, 11640625, 13439556, 18954729, 24078208, 20534697, 30302100
Offset: 1

Views

Author

T. D. Noe, Jan 16 2006

Keywords

Crossrefs

Cf. A115221 (number of 3 X 3 matrices over Z(n) having determinant 0).

Programs

  • Mathematica
    Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]==0, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}, {d, 0, n-1}, {e, 0, n-1}, {f, 0, n-1}]; cnt, {n, 2, 20}]

Formula

a(n)=n^6-A115225(n). For prime n, a(n)=(n^3+n-1)n^2.

A115226 Order of the group of invertible 3 X 3 symmetric matrices over Z(n).

Original entry on oeis.org

1, 28, 468, 1792, 12400, 13104, 100548, 114688, 341172, 347200, 1609300, 838656, 4453488, 2815344, 5803200, 7340032, 22713088, 9552816, 44563284, 22220800, 47056464, 45060400, 141587908, 53673984, 193750000, 124697664, 248714388, 180182016, 574288624, 162489600
Offset: 1

Views

Author

T. D. Noe, Jan 16 2006

Keywords

Comments

Note that A115225 gives the number of 3 x 3 symmetric matrices having nonzero determinant. However, for composite n, a nonzero determinant is not sufficient for the matrix to be invertible; the determinant must also be relatively prime to n.

Links

Crossrefs

Cf. A000056 (order of the group SL(2, Z_n)), A064767 (order of the group GL(3, Z_n)), A115225.

Programs

  • Mathematica
    Table[cnt=0; Do[m={{a, b, c}, {b, d, e}, {c, e, f}}; If[Det[m, Modulus->n]>0 && MatrixQ[Inverse[m, Modulus->n]], cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}, {d, 0, n-1}, {e, 0, n-1}, {f, 0, n-1}]; cnt, {n, 2, 20}]
f[p_, e_] := p^(6*e - 4)*(p^3 - 1)*(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(6*f[i,2] - 4)*(f[i,1]^3 - 1)*(f[i,1] - 1));} \\ Amiram Eldar, Nov 05 2022

Formula

For prime p, a(p) = (p^3-1)*(p-1)*p^2.
In general, a(n) = A115224(n) * phi(n) = A064767(n)/A000056(n).
Multiplicative with a(p^e) = p^(6*e - 4)*(p^3 - 1)*(p - 1). - Amiram Eldar, Sep 10 2020
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^4/((p-1)^3 * (p^2+p+1)^2 * (p^3+1))) = 1.03859354030263389220782701124174403591851545785245128014455467710993780757... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^7, where c = (1/7) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.08230753362... . - Amiram Eldar, Nov 05 2022

Extensions

More terms from Amiram Eldar, Sep 10 2020
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.