\[\begin{array}{|r | r r |} \hline n & n\dagger n+2 & \Delta \\ \hline 0 & 12 & - \\ 1 & 123 & 111 \\ 2 & 234 & 111 \\ 3 & 345 & 111 \\ 4 & 456 & 111 \\ 7 & 789 & - \\ 8 & 8910 & 8121 \\ 9 & 91011 & 82101 \\ 10 & 101112 & 10101 \\ 11 & 111213 & 10101 \\ 97 & 979899 & - \\ 98 & 9899100 & 8919201 \\ 99 & 99100101 & 89201001 \\ 100 & 100101102 & 1001001 \\ ... & ... & - \\ 997 & 997998999 & - \\ 998 & 9989991000 & 8991992001 \\ 999 & 99910001001 & 89920010001 \\ 1000 & - & 100010001 \\ \hline \end{array}\]
This leads to three families of numbers associated with the differences, \[[1|0;k|1|0;k|1], \;\;\; k \in \mathbb{N}^0 \\ [8|9;k|1|9;k|2|0;k|1], \;\;\; k \in \mathbb{N}^0 \\ [8|9;k|2|0;k|1|0;k|01], \;\;\; k \in \mathbb{N}^0\]
Then we have \[\begin{array}{|l|l|} k & [8|9;k|1|9;k|2|0;k|1] \\ \hline 0 & 3×2707 \\ 1 & 3×367×8101 \\ 2 & 3×7331×408857 \\ 3 & 3×61×49175955847 \\ 4 & 3×461×6507534345047 \\ 5 & 3×29×13068901×7915599323 \\ \end{array}\]
It appears there is a trend of always being divisible by three forming for this particular family. In this way we can state a new family relationship to the more fundamental set \[[2|9;k|7|3;k|0|6;k|7]=\frac{1}{3}[8|9;k|1|9;k|2|0;k|1], \;\;\; k \in \mathbb{N}^0,\]
by analysing the form of the ’other’ divisors, that is those not equal to \(3\) in the table above. Then we have \[\begin{array}{|l|l|} ndigs & k & [2|9;k|7|3;k|0|6;k|7] \\ \hline &0 & 2707 \\ &1 & 367×8101 \\ &2 & 7331×408857 \\ &3 & 61×49175955847 \\ &4 & 461×6507534345047 \\ &5 & 29×13068901×7915599323 \\ &6 & 132157×22700271142151431 \\ 25 &7 & 2999999973333333066666667 \\ &8 & 31×134119195669×721553637267553 \\ &9 & 971×3089598351939581187751458977 \\ &10 & 41617×5285509×13638407658308215754239 \\ &11 & 47×2427587751724753×26293503577218173237 \\ &12 & 29×67×191×8083791190283642268886655295467059 \\ &13 & 6823×4922853072028765637×89315947440316415417 \\ &14 & 59×1471×460409×38225179×111669466423×17588499752611651 \\ 49 &15 & 2999999999999999733333333333333306666666666666667 \\ &16 & 97×17378243×1902810619×38565693547×44999108501×538943244589 \\ &17 & 23×1123×854999×208683067618997483×650969803556729644268849819 \\ 58 &18 & 2999999999999999999733333333333333333306666666666666666667 \\ &19 & 43×103×3019×1083463×7470322471885033×27720370278030258074577706955323 \\ &20 & 16253×10657403×52655479033×1327848729329×247710354243298686538325081309 \\ &21 & \ne p \\ &22 & \ne p \\ &23 & \ne p \\ &24 & 29×30643×134967101×25012900528022619660042187099801110590209954791717634851850561 \\ &25 & \ne p \\ &26 & 233×57074713×404869643×557193937725682497848127100766089368113977211817871577285417761 \\ &27 & 59×1523×33386380582481053229019441999324853192665383145813162394767983202940969169532331 \\ &28 & \ne p \\ &29 & \ne p \\ &30 & 643×4665629860031104199066874027993364437532400207361327112493519917055469155002592016588906169 \\ &31 & 509×144203×18071502899×1739144075649997×1300467021504033122697284599677770897073180065778957265077177107 \\ & 32 & \ne p \\ & 33 & \ne p \\ & 34 & \ne p \\ & 35 & \ne p \\ & 36 & \ne p \\ & 37 & \ne p \\ & 38 & \ne p \\ & 39 & \ne p \\ & 40 & 29×43×1447949×5221373×318212193202873202203915859020970384316533247825314047632529269407411811226313804190758990607680705805260093\\ & 41 & \ne p \\ & 42 & 4793×45815352361×13661638669780679124036863023001124578173550024707290545709626024023811674456619238451475241882350377541514421080779\\ & 43 & \ne p \\ & 44 & \ne p \\ & 45 & \ne p \\ & 46 & \ne p \\ %37139×80777619214303023775545922076523331269016397855982480231921520055287792706678514050818097776813233169085507597583851656389958444402559753\\ & 47 & \ne p \\ & 48 & \ne p \\ & 49 & \ne p \\ & 50 & \ne p \\ & 51 & \ne p \\ & 52 & \ne p \\ & 53 & \ne p \\ & 54 & \ne p \\ & 55 & \ne p \\ & 56 & \ne p \\ & 57 & \ne p \\ & 58 & 293×587×17442773168363460878766912222151159072277037751975394061317006897647745133950807503493399848441682025997484364491940468202793557027208788056739403030778742298531124690633037\\ & 59 & \ne p \\ & 60 & 3911×22279601×34429128521134020280530097452556863771517130835644028733975877853802187716818619167678555270320838855615064140652396383384508385807432690210619643741836537573021123186893197\\ & 61 & \ne p \\ \end{array}\]
\[\begin{array}{|l|l|} k & [2|9;k|7|3;k|0|6;k|7] \\ \hline 5 & 29×103448|183907|954023 \\ 12 & 29×103448|2758620|597|7011494252|7816091|954023 \\ 24 & 29×103448|2758620|689655172413|7011494252|873563218390804505747126436|7816091|954023 \\ \end{array}\]
Noting that \(183907\), \(7816091\) are prime. There seems to be no mention of this phrase in the next \(27\) digits section. It is very strange that large non-repeating numbers are occurring.
\[\begin{array}{|l|l|} k & [2|9;k|7|3;k|0|6;k|7] \\ \hline 14 & 59×50847457627118|59887005|6497175|096045197740113 \\ 27 & 59×50847457627118|6440677966101|6497175|1412429378531073446327231638418079|096045197740113 \\ \end{array}\]