

A053696


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


14



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
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OFFSET

1,1


COMMENTS

Numbers of the form (b^n1)/(b1) 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^k1)/(b1) = (b^21)/(b1) = b+1, so every integer m >= 3 is a 2digit repunit in base b = m1. And for n = 1 (the 1digit degenerate repunit) we get (b1)/(b1) = 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


LINKS

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1172 terms from Noe)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avriljuin 2010, pages 3038; included here with permission from the editors of Quadrature.


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


EXAMPLE

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


MAPLE

N:= 10^4: # to get all terms <= N
V:= Vector(N):
for b from 2 while (b^31)/(b1) <= N do
inds:= [seq((b^k1)/(b1), k=3..ilog[b](N*(b1)+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^n1)/(b1); 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 *)


PROG

(PARI) list(lim)=my(v=List(), e, t); for(b=2, sqrt(lim), e=3; while((t=(b^e1)/(b1))<=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^e1)/(b1))<=lim, listput(v, t); e++)); vecsort(concat(Vec(v), vector((sqrtint (lim\1*43)3)\2, i, i^2+3*i+3)), , 8) \\ Charles R Greathouse IV, May 30 2013
(Haskell)
a053696 n = a053696_list !! (n1)
a053696_list = filter ((> 1) . a088323) [2..]
 Reinhard Zumkeller, Jan 22 2014, Nov 26 2013


CROSSREFS

Cf. A090503 (a subsequence), A119598 (numbers that are repunits in four or more bases), A125134, A085104.
Cf. A108348.
Sequence in context: A076196 A167782 A257521 * A090503 A059520 A293576
Adjacent sequences: A053693 A053694 A053695 * A053697 A053698 A053699


KEYWORD

nonn,base,easy


AUTHOR

Henry Bottomley, Mar 23 2000


STATUS

approved



