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A233520
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The number of distinct values of x^x (mod n) - x for x in 0 < x < n.
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4
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0, 1, 2, 2, 4, 2, 5, 4, 5, 5, 6, 4, 10, 7, 8, 9, 11, 5, 12, 9, 12, 10, 15, 9, 14, 12, 14, 12, 19, 11, 21, 19, 18, 16, 19, 12, 28, 18, 18, 18, 30, 13, 33, 20, 22, 23, 36, 18, 28, 20, 23, 27, 39, 17, 35, 24, 32, 30, 43, 20, 46, 33, 26, 37, 37, 22, 49, 34, 34, 30
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OFFSET
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1,3
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COMMENTS
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According to Kurlberg et al. (who quote Crocker and Somer), for primes p, the count is between floor(sqrt((p-1)/2)) and 3p/4 + O(p^(1/2 + o(1))).
Note that the subtraction is not done mod n. - Robert Israel, Dec 17 2014
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LINKS
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EXAMPLE
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For n = 5 the a(5) = 4 values are 1-1=0, 4-2=2, 2-3=-1, 1-4=-3. - Robert Israel, Dec 17 2014
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MAPLE
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f:= n -> nops({seq((x &^ x mod n - x) , x = 1 .. n-1)}):
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MATHEMATICA
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fs[p_] := Module[{x = Range[p - 1]}, Length[Union[PowerMod[x, x, p] - x]]]; Table[fs[n], {n, 100}]
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PROG
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(PARI) a(n) = #Set(vector(n-1, j, lift(Mod(j, n)^j) - j)); \\ Michel Marcus, Dec 16 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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