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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.

%I A276469 #30 Jul 09 2020 07:21:13
%S A276469 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,
%T A276469 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,
%U A276469 1,1,1,1,1,1,1,1,1,1
%N A276469 Triangle read by rows: T(n,k) = n-th cyclotomic polynomial evaluated at x = k and then reduced mod n.
%C A276469 Let C_n(x) denote the n-th cyclotomic polynomial. Then T(n,k) = C_n(k) mod n.
%C A276469 Conjectures:
%C A276469 1) (mod p)  C_p(k) == 1, except C_p(1) == 0, for prime p, 0<=k<p.
%C A276469 2) (mod 2^e) C_[2^e](k) == 1 if k odd, == 0 k even, for e>1, 0<=k<2^e
%C A276469 3) (mod p^e) C_[p^e](k) == 1, except C_[p^e](1+np) = p, e>1, 0<=n<p^(e-1)
%C A276469 4.a) (mod m) C_m(k) for some composite m has values all 1's,
%C A276469      but it is not clear for which m this happens,
%C A276469 4.b) (mod m) C_m(m) for other composite m has values 1 and x,
%C A276469 4.c) with recurring period x
%C A276469 4.d) x is the largest prime dividing m.
%C A276469 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.
%F A276469 T(i,j) = Cyclotomic_i(j) (mod i); for i>=1 and j=0..i-1.
%e A276469 Let C_N(x) be the N'th cyclotomic polynomial, then the values of C_N(k) mod N, m=0,...,N-1, are:
%e A276469     \  0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 -- k -->
%e A276469 C_1:   0
%e A276469 C_2:   1 0
%e A276469 C_3:   1 0 1
%e A276469 C_4:   1 2 1 2
%e A276469 C_5:   1 0 1 1 1
%e A276469 C_6:   1 1 3 1 1 3     (note period 3)
%e A276469 C_7:   1 0 1 1 1 1 1
%e A276469 C_8:   1 2 1 2 1 2 1 2
%e A276469 C_9:   1 3 1 1 3 1 1 3 1     (note period 3)
%e A276469 C_10:  1 1 1 1 5 1 1 1 1 5     (note period 5)
%e A276469 C_11:  1 0 1 1 1 1 1 1 1 1 1
%e A276469 C_12:  1 1 1 1 1 1 1 1 1 1 1 1
%e A276469 C_13:  1 0 1 1 1 1 1 1 1 1 1 1 1
%e A276469 C_14:  1 1 1 1 1 1 7 1 1 1 1 1 1 7     (note period 7)
%e A276469 C_15:  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e A276469 C_16:  1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
%t A276469 Table[Mod[Cyclotomic[i, j], i], {i, 12}, {j, 0, i - 1}] // Flatten (* _Michael De Vlieger_, Sep 23 2016 *)
%o A276469 (PARI) T(n, k) = polcyclo(n, k) % n; \\ _Michel Marcus_, Sep 22 2016
%Y A276469 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).
%K A276469 nonn,tabl
%O A276469 1,8
%A A276469 _Peter A. Lawrence_, Sep 04 2016
%E A276469 a(1) corrected by _Jinyuan Wang_, Jul 09 2020