A206946 Primes of the form (m^p-1)/(m-1), where abs(m) > 1 and p is an odd prime.

3, 7, 11, 13, 31, 43, 61, 73, 127, 157, 211, 241, 307, 421, 463, 521, 547, 601, 683, 757, 1093, 1123, 1483, 1723, 2551, 2731, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9091, 9901, 10303, 11131, 12211, 12433, 13421

Offset: 1

Author

Lei Zhou, Feb 28 2012

Keywords

• nonn

Comments

Old name was: Very generalized repunit prime numbers.

These numbers are prime numbers that can be written in the form of (11...1) base m, with 3 or more ones and |m| > 1.

This sequence is the prime numbers picked from A206943.

Generalized repunit prime numbers are defined in the Caldwell link, as the form of (111...1) base m with numbers of ones >= (1/5)m.

This sequence includes all (111...1) base m primes with m < 1 and number of ones > 2.

Examples

			3 = (111) base -2, so a(1) = 3;
7 = (111) base 2, so a(2) = 7;
11 = (11111) base -2, so a(3) = 11.
31 = (2^5-1)/(2-1) = (5^3-1)/(5-1) = (6^3+1)/(6+1),
43 = (2^7+1)/(2+1) = (7^3+1)/(7+1) = (6^3-1)/(6-1),
8191 = (2^13-1)/(2-1) = (90^3-1)/(90-1) = (91^3+1)/(91+1).
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Chris Caldwell, The Top Twenty: Generalized Repunit

Crossrefs

Cf. A206943, A002275, A066180, A005117.

Programs

Mathematica

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 = 13500; 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] &]; a = {}; Do[i = 0; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a]
nn = 120; ps = Prime[Range[2, PrimePi[Log[2, 2*nn^2 + 1]]]]; t = {}; Do[n = 0; If[Abs[m] > 1, n = (m^p - 1)/(m - 1); If[n > nn^2, n = 0]]; If[PrimeQ[n], AppendTo[t, n]], {p, ps}, {m, -nn, nn}]; t = Union[t] (* T. D. Noe, May 03 2013 *)

Extensions

Better name and new examples by Thomas Ordowski, Apr 28 2013