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A006045
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Sum of orders of all 2 X 2 matrices with entries mod n.
(Formerly M3946)
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2
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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
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OFFSET
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1,2
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COMMENTS
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The order of a matrix M over Z/(nZ) is the smallest k such that M^k is idempotent.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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The article gives an incorrect value for a(5).
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STATUS
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approved
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