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A276469
Triangle read by rows: T(n,k) = n-th cyclotomic polynomial evaluated at x = k and then reduced mod n.
1
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.
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
KEYWORD
nonn,tabl
AUTHOR
Peter A. Lawrence, Sep 04 2016
EXTENSIONS
a(1) corrected by Jinyuan Wang, Jul 09 2020
STATUS
approved