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A164822
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Triangle read by rows, giving the number of solutions mod j of T_k(x) = 1, for j >= 2 and k = 1:j-1, where T_k is the k'th Chebyshev polynomial of the first kind.
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5
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1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 4, 1, 5, 1, 1, 2, 2, 2, 1, 4, 1, 4, 1, 7, 1, 4, 1, 1, 2, 3, 4, 1, 6, 1, 4, 1, 4, 2, 5, 1, 8, 1, 5, 2, 1, 2, 2, 2, 3, 4, 1, 2, 2, 6, 1, 4, 1, 11, 1, 4, 1, 11, 1, 4, 1, 1, 2, 2, 2, 1, 4, 4, 2, 2, 2, 1, 6, 1, 4, 2, 5, 1, 8, 1, 9, 2, 4, 1, 9, 1, 1, 4, 2, 8, 1, 8, 1, 8, 2, 4, 1, 14, 1
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OFFSET
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1,3
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COMMENTS
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T_k(0) = 1 if k == 0 mod 4, but x=0 is not counted as a solution. - Robert Israel, Apr 06 2015
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LINKS
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FORMULA
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a(k,j) is multiplicative in j for each odd k.
a(k,j)+1 is multiplicative in j for k divisible by 4.
a(k,j)+[j=2] is multiplicative in j for k == 2 mod 4, where [j=2] = 1 if j=2, 0 otherwise.
a(1,j) = 1.
a(3,2^m) = 1.
a(3,p^m) = p^floor(m/2)+1 if p is a prime > 3.
a(4,p^m) = p^floor(m/2)+1 if p is a prime > 2.
a(5,p) = 3 if p is in A045468, 1 for other primes p. (End)
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EXAMPLE
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The triangle of numbers is:
.....k..1..2..3..4..5..6..7..8..9.10
..j..
..2.....1
..3.....1..2
..4.....1..2..1
..5.....1..2..2..2
..6.....1..4..1..5..1
..7.....1..2..2..2..1..4
..8.....1..4..1..7..1..4..1
..9.....1..2..3..4..1..6..1..4
.10.....1..4..2..5..1..8..1..5..2
.11.....1..2..2..2..3..4..1..2..2..6
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MAPLE
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seq(seq(nops(select(t -> orthopoly[T](k, t)-1 mod j = 0, [$1..j-1])), k=1..j-1), j=2..20); # Robert Israel, Apr 06 2015
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MATHEMATICA
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Table[Length[Select[Range[j-1], Mod[ChebyshevT[k, #]-1, j] == 0&]], {j, 2, 20}, {k, 1, j-1}] // Flatten (* Jean-François Alcover, Mar 27 2019, after Robert Israel *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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