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A006045 Sum of orders of all 2 X 2 matrices with entries mod n.
(Formerly M3946)
2
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 (list; graph; refs; listen; history; text; internal format)
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)
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
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

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)