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A053696
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Numbers that can be represented as a string of three or more 1's in a base >= 2.
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20
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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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internal format)
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
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FORMULA
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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
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EXAMPLE
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a(5) = 31 because 31 can be written as 111 base 5 (or indeed 11111 base 2).
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
(Haskell)
a053696 n = a053696_list !! (n-1)
a053696_list = filter ((> 1) . a088323) [2..]
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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