A108348 -id:A108348 - cp's OEIS frontend

A053696 Numbers that can be represented as a string of three or more 1's in a base >= 2.

7, 13, 15, 21, 31, 40, 43, 57, 63, 73, 85, 91, 111, 121, 127, 133, 156, 157, 183, 211, 241, 255, 259, 273, 307, 341, 343, 364, 381, 400, 421, 463, 507, 511, 553, 585, 601, 651, 703, 757, 781, 813, 820, 871, 931, 993, 1023, 1057, 1093, 1111, 1123, 1191

Offset: 1

Henry Bottomley, Mar 23 2000

Numbers of the form (b^n-1)/(b-1) for n > 2 and b > 1. - T. D. Noe, Jun 07 2006
Numbers m that are nontrivial repunits for any base b >= 2. For k = 2 (I use k for the exponent since n is used as the index in a(n)) we get (b^k-1)/(b-1) = (b^2-1)/(b-1) = b+1, so every integer m >= 3 is a 2-digit repunit in base b = m-1. And for n = 1 (the 1-digit degenerate repunit) we get (b-1)/(b-1) = 1 for any base b >= 2. If we considered all k >= 1 we would get the sequence of all positive integers except 2 since it is the smallest uniform base used in positional representation (2 might be seen as the "repunit" in a nonpositional base representation such as the Roman numerals where 2 is expressed as II). - Daniel Forgues, Mar 01 2009
These repunits numbers belong to Brazilian numbers (A125134) (see Links: "Les nombres brésiliens" - section IV, p. 32). - Bernard Schott, Dec 18 2012
The Brazilian primes (A085104) belong to this sequence. - Bernard Schott, Dec 18 2012

			a(5) = 31 because 31 can be written as 111 base 5 (or indeed 11111 base 2).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1172 terms from T. D. Noe)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A090503 (a subsequence), A119598 (numbers that are repunits in four or more bases), A125134, A085104.

Cf. A108348.

a053696 n = a053696_list !! (n-1)
a053696_list = filter ((> 1) . a088323) [2..]
-- Reinhard Zumkeller, Jan 22 2014, Nov 26 2013

N:= 10^4: # to get all terms <= N
V:= Vector(N):
for b from 2 while (b^3-1)/(b-1) <= N do
  inds:= [seq((b^k-1)/(b-1), k=3..ilog[b](N*(b-1)+1))];
  V[inds]:= 1;
od:
select(t -> V[t] = 1, [$1..N]); # Robert Israel, Dec 10 2015

fQ[n_] := Block[{d = Rest@ Divisors[n - 1]}, Length@ d > 2 && Length@ Select[ IntegerDigits[n, d], Union@# == {1} &] > 1]; Select[ Range@ 1200, fQ]
lim=1000; Union[Reap[Do[n=3; While[a=(b^n-1)/(b-1); a<=lim, Sow[a]; n++], {b, 2, Floor[Sqrt[lim]]}]][[2, 1]]]
Take[Union[Flatten[With[{l=Table[PadLeft[{},n,1],{n,3,100}]}, Table[ FromDigits[#,n]&/@l,{n,2,100}]]]],80] (* Harvey P. Dale, Oct 06 2011 *)

list(lim)=my(v=List(),e,t);for(b=2,sqrt(lim),e=3;while((t=(b^e-1)/(b-1))<=lim,listput(v,t);e++));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Oct 06 2011

list(lim)=my(v=List(),e,t);for(b=2,lim^(1/3),e=4;while((t=(b^e-1)/(b-1))<=lim,listput(v,t);e++));vecsort(concat(Vec(v), vector((sqrtint (lim\1*4-3)-3)\2,i,i^2+3*i+3)),,8) \\ Charles R Greathouse IV, May 30 2013

a(n) ~ n^2 since as n grows the density of repunits of degree 2 among all the repunits tends to 1. - Daniel Forgues, Dec 09 2008

A088323(a(n)) > 1. - Reinhard Zumkeller, Jan 22 2014

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

Original entry on oeis.org

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

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

nonn

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

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

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

Cf. A053696.

Cf. A000961, A108348.

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

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 *)

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 */

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

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

cp's OEIS Frontend

A053696 Numbers that can be represented as a string of three or more 1's in a base >= 2.

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

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