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

Author

Henry Bottomley, Mar 23 2000

Keywords

• nonn
• base
• easy

Comments

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

Examples

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

Links

• 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.

Crossrefs

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

Cf. A108348.

Programs

Haskell

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

Maple

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

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