[seqfan] Re: functions f:[n]->[n] such that f[(2*x) mod n]=[2*f(x)]

Joerg Arndt arndt at jjj.de
Wed Jun 10 04:28:57 CEST 2009

```* Max Alekseyev <maxale at gmail.com> [Jun 09. 2009 19:12]:
> Joerg,
>
> I've got the following formula for A117987:
>
> [...]
>
> They match your calculations of vector(29,n,yPhi(x^n-1)), except for n=15.
> Could you please check what can be a reason of this disagreement? Is
> it an error somewhere or simply different sequences?
>
> Regards,
> Max
>

Apparently different sequences:

seqs differ for these k<=200:
15, 30, 33, 39, 45, 51, 57, 60, 63, 65, 66, 75, 78, 85, 87, 90, 91,
99, 102, 105, 111, 114, 117, 120, 123, 126, 129, 130, 132, 133, 135,
145, 150, 153, 156, 159, 165, 170, 171, 174, 177, 180, 182, 183, 185,
189, 195, 198,

code used:
\\ *** start code ***
\\ Max Alekseyev:
C(m,d) = sumdiv(d, k, moebius(d/k) * gcd(m,2^k-1)) / d
{ A117987(m) = r=1; fordiv(znorder(Mod(2,m\2^valuation(m,2))),d,
r*=sumdiv(d,q,q*C(m,q))^C(m,d)*2^((2^valuation(m,2)-1)*d*C(m,d))   );  r }
N=200;
t2=vector(N,n,A117987(n));
t1=vector(N,n,yPhi(x^n-1));  \\ as in other mail
dd=t2-t1
for(k=1,#dd,if(dd[k],print1(k", ")))
\\\ *** end code ***

vector dd (differences) starts as
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6750, 0, 0, 0, 0, 0, 0, 0, \
0, 0, 0, 0, 0, 0, 0, 221184000, 0, 0, 71874, 0, 0, 0, 0, 0, 118638, 0, \
0, 0, 0, 0, 123018750, 0, 0, ...]

Numbers in your seq are never smaller than those in mine
(again, for k<200).

Also your numbers appear to be multiples of mine:
? for(k=1,#dd,if(dd[k],print1([k,t2[k]/t1[k]])))
[15, 3][30, 3][33, 3][39, 3][45, 3][51, 9][57, 3][60, 3][63, 63] \
[65, 5][66, 3][75, 9][78, 3][85, 25][87, 3][90, 3][91, 7][99, 3]

```