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%I #38 Nov 27 2014 16:11:36
%S 7,13,15,21,31,40,57,63,73,85,91,121,127,133,156,183,255,273,307,341,
%T 364,381,400,511,553,585,651,757,781,820,871,993,1023,1057,1093,1365,
%U 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
%N Number of hyperplanes in a finite projective space (of some dimension d over some finite field of order q).
%C 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.).
%C Subsequence of A053696. - _Hans Havermann_, Nov 21 2013
%D T. Tsuzuki, Finite groups and finite geometries, Cambridge University Press, 1982, p. 73.
%H Max Alekseyev, <a href="/A090503/b090503.txt">Table of n, a(n) for n = 1..1504</a> (contains all terms below 10^8)
%H P. Buser, J. H. Conway, P. Doyle and K.-D. Semmler, <a href="http://www.math.dartmouth.edu/~doyle/docs/drum/drum.pdf">Isospectral domains</a>
%H W. Cherowitzo, <a href="http://www-math.ucdenver.edu/~wcherowi/courses/m6221/pglc1d.html">Finite projective spaces</a>
%H Y. Okada and A. Shudo, <a href="http://dx.doi.org/10.1088/0305-4470/34/30/303">Equivalence between isospectrality and isolength spectrality for a certain class of planar billiard domains</a>, J. Phys. A: Math. Gen. 34 (2001), 5911-5922
%F Numbers of the form (q^(d+1)-1)/(q-1), d>=2, q=p^m with m>=1 and p prime.
%t 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 *)
%o (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 */
%o (Haskell)
%o a090503 n = a090503_list !! (n-1)
%o a090503_list = f [1..] where
%o f (x:xs) = g $ tail a000961_list where
%o g (q:pps) = h 0 $ map ((`div` (q - 1)) . subtract 1) $
%o iterate (* q) (q ^ 3) where
%o h i (qy:ppys) | qy > x = if i == 0 then f xs else g pps
%o | qy < x = h 1 ppys
%o | otherwise = x : f xs
%o -- _Reinhard Zumkeller_, Nov 26 2013
%Y Cf. A053696.
%Y Cf. A000961, A108348.
%K nonn
%O 1,1
%A Olivier Giraud (olivier.giraud(AT)bristol.ac.uk), Feb 01 2004
%E Missing terms provided by _Jean-François Alcover_ and _Wouter Meeussen_; edited by _M. F. Hasler_, Nov 20 2013
%E PARI program and further terms in a b-file added by _Max Alekseyev_, Nov 20 2013
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