A276469 Triangle read by rows: T(n,k) = n-th cyclotomic polynomial evaluated at x = k and then reduced mod n.

0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Peter A. Lawrence, Sep 04 2016

Keywords

• nonn
• tabl

Comments

Let C_n(x) denote the n-th cyclotomic polynomial. Then T(n,k) = C_n(k) mod n.

Conjectures:

1) (mod p) C_p(k) == 1, except C_p(1) == 0, for prime p, 0<=k

2) (mod 2^e) C_[2^e](k) == 1 if k odd, == 0 k even, for e>1, 0<=k<2^e

3) (mod p^e) C_[p^e](k) == 1, except C_[p^e](1+np) = p, e>1, 0<=n

4.a) (mod m) C_m(k) for some composite m has values all 1's,

but it is not clear for which m this happens,

4.b) (mod m) C_m(m) for other composite m has values 1 and x,

4.c) with recurring period x

4.d) x is the largest prime dividing m.

Remarks: (1) is trivial, I suspect (2) and (3) are simple algebra-crunching, (4) seems to be an interesting question. (4) seems to partition the natural numbers into primes union A253235 union A276628.

Examples

			Let C_N(x) be the N'th cyclotomic polynomial, then the values of C_N(k) mod N, m=0,...,N-1, are:
    \  0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 -- k -->
C_1:   0
C_2:   1 0
C_3:   1 0 1
C_4:   1 2 1 2
C_5:   1 0 1 1 1
C_6:   1 1 3 1 1 3     (note period 3)
C_7:   1 0 1 1 1 1 1
C_8:   1 2 1 2 1 2 1 2
C_9:   1 3 1 1 3 1 1 3 1     (note period 3)
C_10:  1 1 1 1 5 1 1 1 1 5     (note period 5)
C_11:  1 0 1 1 1 1 1 1 1 1 1
C_12:  1 1 1 1 1 1 1 1 1 1 1 1
C_13:  1 0 1 1 1 1 1 1 1 1 1 1 1
C_14:  1 1 1 1 1 1 7 1 1 1 1 1 1 7     (note period 7)
C_15:  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
C_16:  1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
		

Crossrefs

Cf. A253235 (numbers m such that T(m,j) are all 1's), A276628 (composites m such that T(m,j) are not all 1's).

Programs

Mathematica

Table[Mod[Cyclotomic[i, j], i], {i, 12}, {j, 0, i - 1}] // Flatten (* Michael De Vlieger, Sep 23 2016 *)

PARI

T(n, k) = polcyclo(n, k) % n; \\ Michel Marcus, Sep 22 2016

Formula

T(i,j) = Cyclotomic_i(j) (mod i); for i>=1 and j=0..i-1.

Extensions

a(1) corrected by Jinyuan Wang, Jul 09 2020

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