[seqfan] Re: nth cyclotomic polynomial values modulo n

[Peter Lawrence]

 > That's a nice triangle - please go ahead and submit it. When you  
have an
 > A-number for it,
 > you might send a follow-up message here so people can look at it.
 >
 > Best regards
 > Neil
 >
 > Neil J. A. Sloane, President, OEIS Foundation.
 > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
 > Also Visiting Scientist, Math. Dept., Rutgers University,  
Piscataway, NJ.
 > Phone: 732 828 6098; home page: http://NeilSloane.com
 > Email: njasloane at gmail.com

Neil,
      it is A276469, awaiting review, and I am wondering if you can
possibly fast-track it, because

it appears to define a simple function that evaluates to either 1
or the largest prime factor of n, which to me is absolutely astounding,
and I'd like some real Number Theorists to take a look at it.

sincerely,
Peter A. Lawrence.


NAME
allocated for Peter A. Lawrence
modulo N values of N'th cyclotomic polynomial, triangle of

DATA
1, 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
begin {
several patterns are apparent by observation
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 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 with 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 largest prime dividing m
(1) is trivial, I suspect (2) and (3) are simple algebra-crunching,
(4) seems to be a genuine conjecture worth a Number Theorist's time.
(4) seems to partition the natural numbers into
     primes union A253235 union A276628
} end Peter A. Lawrence

FORMULA
a(i,j) = Cyclotomic_i(j) (mod i);  i=1,...;  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:  1
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
A253235: numbers n such that this seq A276469(n,j) are all 1's
A276628: numbers n such that this seq A267469(n,j) are not all 1's

KEYWORD
allocated
nonn,changed

AUTHOR
Peter A. Lawrence, Sep 04 2016

STATUS
approved
editing