Factorization of Repunits, Rn (n

repfactors={{},{{11,1}},{{3,1},{37,1}},{{11,1},{101,1}},{{41,1},{271,1}},{{3,1},{7,1},{11,1},{13,1},{37,1}},{{239,1},{4649,1}},{{11,1},{73,1},{101,1},{137,1}},{{3,2},{37,1},{333667,1}},{{11,1},{41,1},{271,1},{9091,1}},{{21649,1},{513239,1}},{{3,1},{7,1},{11,1},{13,1},{37,1},{101,1},{9901,1}},{{53,1},{79,1},{265371653,1}},{{11,1},{239,1},{4649,1},{909091,1}},{{3,1},{31,1},{37,1},{41,1},{271,1},{2906161,1}},{{11,1},{17,1},{73,1},{101,1},{137,1},{5882353,1}},{{2071723,1},{5363222357,1}},{{3,2},{7,1},{11,1},{13,1},{19,1},{37,1},{52579,1},{333667,1}},{{1111111111111111111,1}},{{11,1},{41,1},{101,1},{271,1},{3541,1},{9091,1},{27961,1}},{{3,1},{37,1},{43,1},{239,1},{1933,1},{4649,1},{10838689,1}},{{11,2},{23,1},{4093,1},{8779,1},{21649,1},{513239,1}},{{11111111111111111111111,1}},{{3,1},{7,1},{11,1},{13,1},{37,1},{73,1},{101,1},{137,1},{9901,1},{99990001,1}},{{41,1},{271,1},{21401,1},{25601,1},{182521213001,1}},{{11,1},{53,1},{79,1},{859,1},{265371653,1},{1058313049,1}},{{3,3},{37,1},{757,1},{333667,1},{440334654777631,1}},{{11,1},{29,1},{101,1},{239,1},{281,1},{4649,1},{909091,1},{121499449,1}},{{3191,1},{16763,1},{43037,1},{62003,1},{77843839397,1}},{{3,1},{7,1},{11,1},{13,1},{31,1},{37,1},{41,1},{211,1},{241,1},{271,1},{2161,1},{9091,1},{2906161,1}},{{2791,1},{6943319,1},{57336415063790604359,1}},{{11,1},{17,1},{73,1},{101,1},{137,1},{353,1},{449,1},{641,1},{1409,1},{69857,1},{5882353,1}},{{3,1},{37,1},{67,1},{21649,1},{513239,1},{1344628210313298373,1}},{{11,1},{103,1},{4013,1},{2071723,1},{5363222357,1},{21993833369,1}},{{41,1},{71,1},{239,1},{271,1},{4649,1},{123551,1},{102598800232111471,1}},{{3,2},{7,1},{11,1},{13,1},{19,1},{37,1},{101,1},{9901,1},{52579,1},{333667,1},{999999000001,1}},{{2028119,1},{247629013,1},{2212394296770203368013,1}},{{11,1},{909090909090909091,1},{1111111111111111111,1}},{{3,1},{37,1},{53,1},{79,1},{265371653,1},{900900900900990990990991,1}},{{11,1},{41,1},{73,1},{101,1},{137,1},{271,1},{3541,1},{9091,1},{27961,1},{1676321,1},{5964848081,1}},{{83,1},{1231,1},{538987,1},{201763709900322803748657942361,1}},{{3,1},{7,2},{11,1},{13,1},{37,1},{43,1},{127,1},{239,1},{1933,1},{2689,1},{4649,1},{459691,1},{909091,1},{10838689,1}},{{173,1},{1527791,1},{1963506722254397,1},{2140992015395526641,1}},{{11,2},{23,1},{89,1},{101,1},{4093,1},{8779,1},{21649,1},{513239,1},{1052788969,1},{1056689261,1}},{{3,2},{31,1},{37,1},{41,1},{271,1},{238681,1},{333667,1},{2906161,1},{4185502830133110721,1}},{{11,1},{47,1},{139,1},{2531,1},{549797184491917,1},{11111111111111111111111,1}},{{35121409,1},{316362908763458525001406154038726382279,1}},{{3,1},{7,1},{11,1},{13,1},{17,1},{37,1},{73,1},{101,1},{137,1},{9901,1},{5882353,1},{99990001,1},{9999999900000001,1}},{{239,1},{4649,1},{505885997,1},{1976730144598190963568023014679333,1}},{{11,1},{41,1},{251,1},{271,1},{5051,1},{9091,1},{21401,1},{25601,1},{182521213001,1},{78875943472201,1}},{{3,1},{37,1},{613,1},{210631,1},{2071723,1},{52986961,1},{5363222357,1},{13168164561429877,1}},{{11,1},{53,1},{79,1},{101,1},{521,1},{859,1},{265371653,1},{1058313049,1},{1900381976777332243781,1}},{{107,1},{1659431,1},{1325815267337711173,1},{47198858799491425660200071,1}},{{3,3},{7,1},{11,1},{13,1},{19,1},{37,1},{757,1},{52579,1},{333667,1},{70541929,1},{14175966169,1},{440334654777631,1}},{{41,1},{271,1},{1321,1},{21649,1},{62921,1},{513239,1},{83251631,1},{1300635692678058358830121,1}},{{11,1},{29,1},{73,1},{101,1},{137,1},{239,1},{281,1},{4649,1},{7841,1},{909091,1},{121499449,1},{127522001020150503761,1}},{{3,1},{37,1},{21319,1},{10749631,1},{1111111111111111111,1},{3931123022305129377976519,1}},{{11,1},{59,1},{3191,1},{16763,1},{43037,1},{62003,1},{77843839397,1},{154083204930662557781201849,1}},{{2559647034361,1},{4340876285657460212144534289928559826755746751,1}},{{3,1},{7,1},{11,1},{13,1},{31,1},{37,1},{41,1},{61,1},{101,1},{211,1},{241,1},{271,1},{2161,1},{3541,1},{9091,1},{9901,1},{27961,1},{2906161,1},{4188901,1},{39526741,1}},{{733,1},{4637,1},{329401,1},{974293,1},{1360682471,1},{106007173861643,1},{7061709990156159479,1}},{{11,1},{2791,1},{6943319,1},{57336415063790604359,1},{909090909090909090909090909091,1}},{{3,2},{37,1},{43,1},{239,1},{1933,1},{4649,1},{10837,1},{23311,1},{45613,1},{333667,1},{10838689,1},{45121231,1},{1921436048294281,1}},{{11,1},{17,1},{73,1},{101,1},{137,1},{353,1},{449,1},{641,1},{1409,1},{19841,1},{69857,1},{976193,1},{5882353,1},{6187457,1},{834427406578561,1}},{{41,1},{53,1},{79,1},{271,1},{265371653,1},{162503518711,1},{5538396997364024056286510640780600481,1}},{{3,1},{7,1},{11,2},{13,1},{23,1},{37,1},{67,1},{4093,1},{8779,1},{21649,1},{513239,1},{599144041,1},{183411838171,1},{1344628210313298373,1}},{{493121,1},{79863595778924342083,1},{28213380943176667001263153660999177245677,1}},{{11,1},{101,1},{103,1},{4013,1},{2071723,1},{28559389,1},{1491383821,1},{5363222357,1},{21993833369,1},{2324557465671829,1}},{{3,1},{37,1},{277,1},{203864078068831,1},{11111111111111111111111,1},{1595352086329224644348978893,1}},{{11,1},{41,1},{71,1},{2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Length[repfactors]

352

...that is, from FactorInteger[R1], which Mathematica equates to {}, to FactorInteger[R352].

Union[Table[tr=Transpose[repfactors[[i]]];Apply[Times,First[tr]^Last[tr]]==(10^i-1)/9,{i,2,Length[repfactors]}]]
composites=Flatten[Position[Table[Count[PrimeQ[First[Transpose[repfactors[[i]]]]],False],{i,1,Length[repfactors]}],1]]

{True}
{}