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A090503
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Number of hyperplanes in a finite projective space (of some dimension d over some finite field of order q).
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4
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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
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listen;
history;
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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T. Tsuzuki, Finite groups and finite geometries, Cambridge University Press, 1982, p. 73.
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LINKS
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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
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FORMULA
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Numbers of the form (q^(d+1)-1)/(q-1), d>=2, q=p^m with m>=1 and p prime.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A053696.
Cf. A000961, A108348.
Sequence in context: A326380 A257521 A053696 * A059520 A293576 A233301
Adjacent sequences: A090500 A090501 A090502 * A090504 A090505 A090506
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KEYWORD
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nonn
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AUTHOR
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Olivier Giraud (olivier.giraud(AT)bristol.ac.uk), Feb 01 2004
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EXTENSIONS
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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
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STATUS
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approved
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