A006045 Sum of orders of all 2 X 2 matrices with entries mod n. (Formerly M3946)

1, 26, 272, 722, 5270, 5260, 37358, 18414, 56216, 95668, 487714, 99796, 1304262, 627046, 593398, 481982, 7044222, 931396, 11570384, 1602940, 4037650, 8694134, 40220524, 2069292, 15855230, 21686124, 13215872, 10948486, 129952894, 10451648

Offset

1,2

Comments

The order of a matrix M over Z/(nZ) is the smallest k such that M^k is idempotent.

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..150 (first 61 terms from Sean A. Irvine)

Michael S. Branicky, Python program

A. Wilansky, Spectral decomposition of matrices for high school students, Math. Mag., vol. 41, 1968, pp. 51-59.

A. Wilansky, Spectral decomposition of matrices for high school students, Math. Mag., vol. 41, 1968, pp. 51-59. (Annotated scanned copy)

A. Wilansky, Letters to N. J. A. Sloane, Jun. 1991.

Prog

(PARI) order(m) = {kk = 1; ok = 0; while (! ok, mk = m^kk; if (mk^2 == mk, ok = 1, kk++); ); return(kk); }
a(n) = {ret = 0; m = matrix(2, 2); for (i=0, n-1, m[1, 1] = Mod(i, n); for (j=0, n-1, m[1, 2] = Mod(j, n); for (k=0, n-1, m[2, 1] = Mod(k, n); for (l=0, n-1, m[2, 2] = Mod(l, n); ret += order(m); ); ); ); ); return (ret); }
(Python) # see link for faster version
from itertools import product
def mmm2(A, B, modder): # matrix multiply modulo for 2x2
  return ((A[0]*B[0]+A[1]*B[2])%modder, (A[0]*B[1]+A[1]*B[3])%modder,
          (A[2]*B[0]+A[3]*B[2])%modder, (A[2]*B[1]+A[3]*B[3])%modder)
def order(A, modder):
  Ak, k = A, 1
  while mmm2(Ak, Ak, modder) != Ak: Ak, k = mmm2(Ak, A, modder), k+1
  return k
def a(n): return sum(order(A, n) for A in product(range(n), repeat=4))
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, Jan 26 2021

Crossrefs

Sequence in context: A328874 A195755 A186261 * A022686 A200555 A130901

Adjacent sequences: A006042 A006043 A006044 * A006046 A006047 A006048

Keyword

nonn

Author

N. J. A. Sloane, Albert Wilansky

Extensions

The article gives an incorrect value for a(5).

More terms from Michel Marcus, Jun 07 2013

More terms from Sean A. Irvine, Dec 18 2016

Status

approved