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
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