A090503 Number of hyperplanes in a finite projective space (of some dimension d over some finite field of order q).

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

Offset: 1

Author

Olivier Giraud (olivier.giraud(AT)bristol.ac.uk), Feb 01 2004

Keywords

• nonn

Comments

The number of tiles building the known pairs of Euclidean isospectral billiards are 7, 13, 15, 21, ... (see Refs Okada et al. and Buser et al.).

Subsequence of A053696. - Hans Havermann, Nov 21 2013

References

• T. Tsuzuki, Finite groups and finite geometries, Cambridge University Press, 1982, p. 73.

Links

• Max Alekseyev, Table of n, a(n) for n = 1..1504 (contains all terms below 10^8)
• P. Buser, J. H. Conway, P. Doyle and K.-D. Semmler, Isospectral domains
• W. Cherowitzo, Finite projective spaces
• Y. Okada and A. Shudo, Equivalence between isospectrality and isolength spectrality for a certain class of planar billiard domains, J. Phys. A: Math. Gen. 34 (2001), 5911-5922

Crossrefs

Cf. A053696.

Cf. A000961, A108348.

Programs

Haskell

a090503 n = a090503_list !! (n-1)
a090503_list = f [1..] where
   f (x:xs) = g $ tail a000961_list where
     g (q:pps) = h 0 $ map ((`div` (q - 1)) . subtract 1) $
                           iterate (* q) (q ^ 3) where
       h i (qy:ppys) | qy > x    = if i == 0 then f xs else g pps
                     | qy < x    = h 1 ppys
                     | otherwise = x : f xs
-- Reinhard Zumkeller, Nov 26 2013

Mathematica

isA090503[n_] := Module[{f = FactorInteger[n-1]}, For[i = 1, i <= Length[f], i++, For[j = 1, j <= f[[i, 2]], j++, q = f[[i, 1]]^j; If[q == n-1, Continue[]]; If[n*(q-1)+1 == q^IntegerExponent[n*(q-1)+1, q], Return[True]]]]; False]; Reap[For[n = 2, n <= 10^5, n++, If[isA090503[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Nov 21 2013, translated and adapted from Max Alekseyev's program *)

PARI

isA090503(n) = my(f,q); 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 /* Max Alekseyev, Nov 20 2013 */

Formula

Numbers of the form (q^(d+1)-1)/(q-1), d>=2, q=p^m with m>=1 and p prime.

Extensions

Missing terms provided by Jean-François Alcover and Wouter Meeussen; edited by M. F. Hasler, Nov 20 2013

PARI program and further terms in a b-file added by Max Alekseyev, Nov 20 2013