A115224 -id:A115224 - cp's OEIS frontend

A381680 Euler transform of A115224.

1, 1, 29, 263, 1565, 11217, 74412, 482638, 2987123, 18066149, 107415185, 623612637, 3552605428, 19882256022, 109518424910, 594290145192, 3179607733480, 16790129919934, 87573088547032, 451477766533886, 2302069862201553, 11616226357007259, 58036597014533469

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Cf. A061255, A381679.

Cf. A115224, A084220.

a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[6, k^2]/DivisorSigma[3, k^2]*a[n-k], {k, 1, n}]/n; Table[a[n], {n, 0, 30}] (* Vaclav Kotesovec, Mar 04 2025 *)

my(N=30, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 6)/sigma(k^2, 3)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^A115224(k).
G.f.: exp( Sum_{k>=1} sigma_6(k^2)/sigma_3(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_6(k^2)/sigma_3(k^2) * a(n-k).
log(a(n)) ~ 7 * 5^(2/7) * zeta(7)^(1/7) * n^(6/7) / (2^(2/7) * 3^(3/7) * Pi^(4/7)). - Vaclav Kotesovec, Mar 04 2025

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

T. D. Noe, Jan 16 2006

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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 *)

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

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

More terms from Amiram Eldar, Sep 10 2020

cp's OEIS Frontend

A381680 Euler transform of A115224.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula