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A090503 Number of hyperplanes in a finite projective space (of some dimension d over some finite field of order q). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

FORMULA

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

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 *)

PROG

(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 */

(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

CROSSREFS

Cf. A053696.

Cf. A000961, A108348.

Sequence in context: A326380 A257521 A053696 * A059520 A293576 A233301

Adjacent sequences: A090500 A090501 A090502 * A090504 A090505 A090506

KEYWORD

nonn

AUTHOR

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

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

STATUS

approved

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Last modified January 28 10:39 EST 2023. Contains 359859 sequences. (Running on oeis4.)