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A195812
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Sum of the distinct residues of x^n (mod n), x=0..n-1.
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5
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0, 1, 3, 1, 10, 8, 21, 1, 9, 25, 55, 14, 78, 42, 105, 1, 136, 20, 171, 22, 84, 110, 253, 26, 50, 169, 27, 84, 406, 150, 465, 1, 528, 289, 595, 38, 666, 342, 273, 42, 820, 130, 903, 198, 315, 460, 1081, 50, 147, 125, 1275, 156, 1378, 56, 385, 140, 570, 841
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OFFSET
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1,3
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COMMENTS
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a(n) has the following properties :
If n is a power of 2 then a(n)= 1 ;
Except for n = 9 where a(9)=9, if a(n) is the square of a prime p, the sequence shows that n is of the form n = 2p.
The numbers m such that a(m) are square are : 1, 2, 4, 8, 9, 10, 16, 26, 32, 34, 58, 64, 74, 81, ...
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LINKS
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EXAMPLE
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a(10) = 25 because the residues (mod 10) of x^10 are 0, 1, 4, 5, 6, 9 and the sum 25 is a square => a(10) = a(2*5)= 5^2.
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MAPLE
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sumDistRes := proc(n)
local re, x, r ;
re := {} ;
for x from 0 to n-1 do
re := re union { modp(x^n, n) } ;
end do:
add(r, r=re) ;
end proc:
for n from 1 to 100 do
printf("%d, ", sumDistRes(n));
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MATHEMATICA
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Table[{m, Total[Union[Table[PowerMod[x, m, m], {x, m-1}]]]}, {m, 1000}] (* Zak Seidov, Oct 06 2011 *)
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PROG
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(PARI) a(n) = vecsum(Set(vector(n, k, lift(Mod(k-1, n)^n)))); \\ Michel Marcus, Jun 01 2015
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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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