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A206943
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Generalized repeat unit one numbers: all numbers of the form (m^p-1)/(m-1), where abs(m) > 1 and p is odd prime.
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1
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3, 7, 11, 13, 21, 31, 43, 57, 61, 73, 91, 111, 121, 127, 133, 157, 183, 205, 211, 241, 273, 307, 341, 343, 381, 421, 463, 507, 521, 547, 553, 601, 651, 683, 703, 757, 781, 813, 871, 931, 993, 1057, 1093, 1111, 1123, 1191, 1261, 1333, 1407, 1483, 1555, 1561
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OFFSET
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1,1
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COMMENTS
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Here we define "generalized repeat unit one numbers" as numbers that can be represented in the form 11...1_m where the number of ones is k > 2 and |m| > 1.
Normal repeat unit one numbers (a.k.a. "repunits") are numbers in the form 11...1_10 with 2 or more ones.
Trivially, any number n = 11_(n-1).
These numbers take the form of cyclotomic polynomial number Phi(k,m) with k in the form 2^i*p^j, where p is prime and i >= 0, j >= 1. It has p digits of one base -m^(2^(i-1)*p^(j-1)) when i > 0 or base m^(p^(j-1)) when i = 0.
This sequence is a subsequence of A206942.
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LINKS
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EXAMPLE
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111_(-2) = 3, so 3 is a term;
111_2 = 7, so 7 is a term;
11111_(-2) = 11, so 11 is a term.
3 = (2^3 + 1)/(2 + 1);
7 = (2^3 - 1)/(2 - 1) = (3^3 + 1)/(3 + 1);
11 = (2^5 + 1)/(2 + 1).
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MATHEMATICA
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phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; maxdata = 1600; max = Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/4]*2; eb = 2*Floor[(Log[2, maxdata])/2 + 0.5]; While[eg = phiinv[eb]; lu = Length[eg]; lu == 0, eb = eb + 2]; t = Select[Range[eg[[Length[eg]]]], ((EulerPhi[#] <= eb) && (a = FactorInteger[#]; b = Length[a]; (((b == 1) && (a[[1]][[1]] > 2)) || ((b == 2) && (a[[1]][[1]] == 2))))) &]; ap = SortBy[t, Cyclotomic[#, 2] &]; an = SortBy[t, Cyclotomic[#, -2] &]; a = {}; Do[i = 0; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata, a = Append[a, cc]]; i = 0; While[i++; cc = Cyclotomic[an[[i]], -m]; cc <= maxdata, a = Append[a, cc]], {m, 2, max}]; Union[a]
nn = 40; ps = Prime[Range[2, PrimePi[Log[2, 2*nn^2 + 1]]]]; t = {}; Do[If[Abs[m] > 1, n = (m^p - 1)/(m - 1); If[n < nn^2, AppendTo[t, n]]], {p, ps}, {m, -nn, nn}]; t = Union[t] (* T. D. Noe, May 03 2013 *)
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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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EXTENSIONS
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STATUS
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approved
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