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A164822 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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T_k(0) = 1 if k == 0 mod 4, but x=0 is not counted as a solution. - Robert Israel, Apr 06 2015

LINKS

C. H. Gribble, Flattened triangle, for j = 2:100 and k = 1:j-1.

FORMULA

From Robert Israel, Apr 06 2015 (Start):

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(2,j) = A060594(j) if j is odd, A060594(j/2) if j is even.

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)

EXAMPLE

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

MAPLE

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

MATHEMATICA

Table[Length[Select[Range[j-1], Mod[ChebyshevT[k, #]-1, j] == 0&]], {j, 2, 20}, {k, 1, j-1}] // Flatten (* Jean-Fran├žois Alcover, Mar 27 2019, after Robert Israel *)

CROSSREFS

Cf. A045468, A164823, A164831, A164846, A165252.

Sequence in context: A307322 A306737 A178474 * A110627 A205107 A228667

Adjacent sequences:  A164819 A164820 A164821 * A164823 A164824 A164825

KEYWORD

nonn,tabl

AUTHOR

Christopher Hunt Gribble, Aug 27 2009

EXTENSIONS

Sequence and definition corrected by Christopher Hunt Gribble, Sep 10 2009

Minor edit by N. J. A. Sloane, Sep 13 2009

STATUS

approved

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Last modified October 22 22:35 EDT 2021. Contains 348180 sequences. (Running on oeis4.)