# [seqfan] Re: About A090503 Number of hyperplanes

Max Alekseyev maxale at gmail.com
Wed Nov 20 20:18:56 CET 2013

```It is easy to see that (q^(d+1)-1)/(q-1) == 1 (mod q), implying that
testing whether n belongs to this sequence can done by testing all
prime powers q dividing n-1.

Here are all terms below 10^5 (confirming terms computed by Wouter):

{ isA090503(n) = f=factor(n-1); for(i=1,matsize(f)[1], for(j=1,f[i,2],
q=f[i,1]^j; if(q==n-1,next); if( n*(q-1)+1 ==
q^valuation(n*(q-1)+1,q), return(q)); )); 0 }

? for(n=2,10^5, if(isA090503(n),print1(n,", ")) )
7, 13, 15, 21, 31, 40, 57, 63, 73, 85, 91, 121, 127, 133, 156, 183,
255, 273, 307, 341, 364, 381, 400, 511, 553, 585, 651, 757, 781, 820,
871, 993, 1023, 1057, 1093, 1365, 1407, 1464, 1723, 1893, 2047, 2257,
2380, 2451, 2801, 2863, 3280, 3541, 3783, 3906, 4095, 4161, 4369,
4557, 4681, 5113, 5220, 5403, 5461, 6321, 6643, 6973, 7240, 7381,
8011, 8191, 9507, 9841, 10303, 10713, 11557, 11991, 12720, 12883,
14763, 15751, 16105, 16257, 16276, 16383, 16513, 17293, 18907, 19461,
19531, 19608, 20440, 21845, 22351, 22953, 24807, 25260, 26733, 28057,
28731, 29524, 30103, 30784, 30941, 32221, 32767, 32943, 33825, 36673,
37443, 37449, 39007, 39801, 44733, 49953, 51757, 52060, 52671, 54523,
57361, 58323, 59293, 63253, 65535, 65793, 66307, 66430, 69433, 69905,
70644, 72631, 73713, 77007, 79243, 80373, 81400, 83811, 86143, 87381,
88573, 88741, 94557, 97033, 97656, 98283,

Regards,
Max

On Wed, Nov 20, 2013 at 11:29 AM, Wouter Meeussen
<wouter.meeussen at telenet.be> wrote:
> I can confirm Jean-François' results.
>
> It seems like the original submision did not extend the range of 'd' far
> enough.
>
> The links in A090503 are 'pining for the fjords' and so quite unresponsive.
>
> A slightly more extended search:
>
> uni = Sort@ Flatten at Table[((p^m)^(d + 1) - 1)/(p^m - 1), {d, 2, 36}, {m, 1,
>      16}, {p, Prime /@ Range[999]}];
> Select[Range[5000], MemberQ[uni, #] &]
>
> {7,13,15,21,31,40,57,63,73,85,91,121,127,133,156,183,255,273,307,341,364,
> 381,400,511,553,585,651,757,781,820,871,993,1023,1057,1093,1365,1407,
> 1464,1723,1893,2047,2257,2380,2451,2801,2863,3280,3541,3783,3906,4095,
> 4161,4369,4557,4681}
>
>
> Wouter.
>
> -----Original Message----- From: Neil Sloane
> Sent: Wednesday, November 20, 2013 4:34 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: About A090503 Number of hyperplanes
>
>
> Well, I do know something about the subject, and the
> formula:
>
> Numbers of the form (q^(d+1)-1)/(q-1), d>=2, q=p^m with m>=1 and p prime
>
> is correct! Compute all numbers of that form,
> remove duplicates, and sort them into increasing order.
>
> But I don't know much about Mathematica, so I can't check your program. If
> it is doing
> what the formula says, then your version is correct,
> and A090503 should be changed.
>
> I wonder if the incorrect sequence is given in any of the references? If so
> we might
> make a new entry for the wrong sequence,
> since the OEIS policy is to include published sequences that are wrong
> (giving a pointer to the correct entry).
>
> Neil
>
>
>
>
> On Wed, Nov 20, 2013 at 10:16 AM, Jean-François Alcover <
> jf.alcover at gmail.com> wrote:
>
>> Dear SeqFans,
>>
>> While checking A090503 (without knowing anything about the subject)
>> I spotted missing terms, assuming the formula meant "All numbers of the
>> form".
>> For instance 127 = ((2^1)^(6 + 1) - 1)/(2^1 - 1) is missing,
>>        though 63 = ((2^1)^(5 + 1) - 1)/(2^1 - 1) is present.
>> Same trouble with 255, 511, 1023, 1093, 2047, 3280, 4095, ...
>> I presume the formula actually meant "Some numbers of the form".
>> Any idea?
>>
>> J.-F. Alcover
>>
>> p.s. This is the way I did it with Mma:
>>
>> f[n_] := Do[If[n == ( (p^m)^(d + 1) - 1)/( p^m - 1), Print[{n, d, m, p}];
>>    Return[True]], {d, 2, 11}, {m, 1, 6}, {p, Prime /@ Range[18]}]
>>
>>  Select[Range[200], f[#] === True &]
>>
>>  {7,2,1,2}
>>
>>  {13,2,1,3}
>>
>>  {15,3,1,2}
>>
>>  {21,2,2,2}
>>
>>  {31,2,1,5}
>>
>>  {40,3,1,3}
>>
>>  {57,2,1,7}
>>
>>  {63,5,1,2}
>>
>>  {73,2,3,2}
>>
>>  {85,3,2,2}
>>
>>  {91,2,2,3}
>>
>>  {121,4,1,3}
>>
>>  {127,6,1,2}
>>
>>  {133,2,1,11}
>>
>>  {156,3,1,5}
>>
>>  {183,2,1,13}
>>
>>  {7, 13, 15, 21, 31, 40, 57, 63, 73, 85, 91, 121, 127, 133, 156, 183}
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> 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.