[seqfan] Re: nth cyclotomic polynomial values modulo n
Neil Sloane
njasloane at gmail.com
Thu Sep 8 20:32:24 CEST 2016
Peter, The definition of A276469 seems a bit unclear!
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
On Thu, Sep 8, 2016 at 11:55 AM, Peter Lawrence <peterl95124 at sbcglobal.net>
wrote:
> > 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
> <http://list.seqfan.eu/cgi-bin/mailman/listinfo/seqfan>
>
> 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* <https://oeis.org/A253235>* union **A276628*
> <https://oeis.org/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* <https://oeis.org/A253235>*: numbers n such that this seq *
> *A276469* <https://oeis.org/A276469>*(n,j) are all 1's*
> *A276628* <https://oeis.org/A276628>*: numbers n such that this seq *
> *A267469* <https://oeis.org/A267469>*(n,j) are not all 1's*
>
> KEYWORD
> allocated
> nonn*,*changed
>
> AUTHOR
> *Peter A. Lawrence* <https://oeis.org/wiki/User:Peter_A._Lawrence>*, Sep
> 04 2016*
>
> STATUS
> approved
> *editing*
>
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