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

1,8

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<p.

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<p^(e-1)

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.

Links

Table of n, a(n) for n=1..78.

Formula

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

Example

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

Mathematica

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

Prog

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

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).

Sequence in context: A221167 A286134 A336922 * A272356 A102565 A076826

Adjacent sequences: A276466 A276467 A276468 * A276470 A276471 A276472

Keyword

nonn,tabl

Author

Peter A. Lawrence, Sep 04 2016

Extensions

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

Status

approved