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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) 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*B+A*B)%modder, (A*B+A*B)%modder,           (A*B+A*B)%modder, (A*B+A*B)%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

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Last modified June 28 02:50 EDT 2022. Contains 354903 sequences. (Running on oeis4.)