Introduce the transform \(T\) on a number \(N\), in base \(10\). Say, \(N=3024\), then the unique prime representation is \(2^4×3^3×7\), and we can write a new number \(T(3024)=22223337\). Repeating, we have \(3×61×121439\) and \(T^2(3024)=361121439\), again \(3×7×43×399913\), giving \(T^3(3024)=3743399913\). etc.
\[\begin{array}{|c|c|c|c|c|c|C|} \hline m & N & T(N) & T^2(N) & T^3(N) & T^4(N) & T^5(N) & T^6(N) & T^7(N) & T^8(N) & T^9(N) & T^{10}(N) & T^{11}(N) & T^{12}(N) & T^{13}(N) & T^{14}(N) & T^{15}(N) & T^{16}(N) & T^{17}(N)\\ \hline \infty & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 3 & 3 \\ 2 & 4 & 2|2 & 2|11 & 211\\ 0 & 5 & 5 \\ 1 & 6 & 2|3 & 23\\ 0 & 7 & 7 \\ 13 & 8 & 2|2|2 & 2|3|3|7 & 3|19|41 & 3|3|3|7|13|13 & 3|11123771 & 7|149|317|941 & 229|31219729 & 11|2084656339 & 3|347|911|118189 & 11|613|496501723 & 97|130517|917327 & 53|1832651281459 & 3|3|3|11|139|653|3863|5107 & 3331113965338635107\\ 2 & 9 & 3|3 & 3|11 & 311 \\ 4 & 10 & 2|5 & 5|5 & 5|11 & 7|73 & 773 \\ 0 & 11 & 11 \\ 1 & 12 & 2|2|3 & 223 \\ 0 & 13 & 13 \\ 5 & 14 & 2|7 & 3|3|3 & 3|3|37 & 47|71 & 13|367 & 13367 \\ 4 & 15 & 3|5 & 5|7 & 3|19 & 11|29 & 1129 \\ 4 & 16 & 2|2|2|2 & 2|11|101 & 3|11|6397 & 3|163|6373 & 31636373 \\ 0 & 17 & 17 \\ 1 & 18 & 2|3|3 & 233 \\ 0 & 19 & 19 \\ 15 & 20 & 2|2|5 & 3|3|5|5 & 5|11|61 & 11|4651 & 3|3|12739 & 17|194867 & 19|41|22073 & 709|273797 & 3|97|137|17791 & 11|3610337981 & 7|3391|4786213 & 3|3|3|3|7|23|31|1815403 & 13|17|23|655857429041 & 7|7|2688237874641409 & 3|31|8308475676071413 & 3318308475676071413\\ 1 & 21 & 3|7 & 37 \\ 1 & 22 & 2|11 & 211 \\ 0 & 23 & 23 \\ 2 & 24 & 2|2|2|3 & 3|3|13|19 & 331319 \\ 3 & 25 & 5|5 & 5|11 & 7|73 & 773 \\ 4 & 26 & 2|13 & 3|71 & 7|53 & 3|251 & 3251 \\ 4 & 27 & 3|3|3 & 3|3|37 & 47|71 & 13|367 & 13367 \\ 1 & 28 & 2|2|7 & 227 \\ 0 & 29 & 29 \\ 2 & 30 & 2|3|5 & 5|47 & 547 \\ 0 & 31 & 31 \\ 2 & 32 & 2|2|2|2|2 & 2|41|271 & 241271 \\ 1 & 33 & 3|11 & 311 \\ 16 & 34 & 2|17 & 7|31 & 17|43 & 3|7|83 & 3|13|97 & 3|29|131 & 11|29921 & 13|23|3779 & 13|433|2351 & 17|23|343561 & 37|46576853 & 3|3|416286317 & 17|1965663901 & 3|3|3|6369098663 & 2897|1151663479 & 3|397|743|2339|13997 & 3397743233913997 \\ 3 & 35 & 5|7 & 3|19 & 11|29 & 1129 \\ 2 & 36 & 2|2|3|3 & 7|11|29 & 71129 \\ 0 & 37 & 37 \\ 2 & 38 & 2|19 & 3|73 & 373 \\ 1 & 39 & 3|13 & 313 \\ 9 & 40 & 2|2|2|5 & 5|5|89 & 3|3|3|3|3|23 & 7|7|59|1153 & 29|2675557 & 3|31|3147049 & 809|1019|4019 & 3|53639|502807 & 3|31|41|92745739 & 3314192745739 \\ 0 & 41 & 41 \\ 2 & 42 & 2|3|7 & 3|79 & 379 \\ 0 & 43 & 43 \\ 9 & 44 & 2|2|11 & 3|11|67 & 3|3|3463 & 13|113|227 & 173|229|331 & 11|15748121 & 541|2062381 & 11|607|810553 & 2281|5088913 & 22815088913 \\ 6 & 45 & 3|3|5 & 5|67 & 3|3|3|3|7 & 17|37|53 & 239|727 & 3|41|1949 & 3411949 \\ 1 & 46 & 2|23 & 223 \\ 0 & 47 & 47 \\ 15 & 48 & 2|2|2|2|3 & 71|313 & 3|11|2161 & 3|13|199|401 & 19|43|109|3517 & 11|17|109|877|1087 & 23|1481|7039|46591 & 3|3|7|53|67|1034726207 & 3|11251223678242069 & 23|4583|2952795526741 & 359|5782291|1130063089 & 835996339|43011938251 & 31|49123|54898161457127 & 467|79367|8496358995643 & 61|61|79|1591356884791277 & 6161791591356884791277 \\ - & 49 & 7|7 & 7|11 & 3|3|79 & 31|109 & 13|2393 & 3|44131 & 17|31|653 & 7|11|43|523 & 11|11|5771019 & 311|35742029 & 7|17|261644891 & 11|19|3431873899 & 11|613|4799|345907 & 3|204751|189066719 & 3|1068250396355573 & 621611|49980213343 & 3|3|6906794442245927 & 73|4615161567701999 & 3|13|18836286194043641 & 3|3|3|43|14369|161461|11627309 & 3|32057|1618455677|2142207827 & 3|1367|2221|5573|475297|1376323127 & 7|3391|51263|25777821480557336017 & 47|67|347|431|120361987|12947236602187 & 3|7|7|17|12809|57470909|57713323|4490256751 & 3096049809383|121823389214993262890297 & 73796236325118712936424989555929478399 & 13|1181|145261411|33089538087518197265265053 & 3|19|521|441731977174163487542111577539726749 & 59|5415617656474189392601483764603009147911 & 13|8423|1466957|3706744784027901056001426046777 & 3|12919|2501509379|96709539317201|1476342474406759 & 3|2039|2713|3121|399320591|151296378525102203388346189 & 13|3119|651853|9121952491|13288820301002347322382772769\\ 2 & 50 & 2|5|5 & 3|5|17 & 3517 \\ 1 & 51 & 3|17 & 317 \\ 1 & 52 & 2|2|13 & 2213 \\ 0 & 53 & 53 \\ 1 & 54 & 2|3|3|3 & 2333 \\ 2 & 55 & 5|11 & 7|73 & 773 \\ 3 & 56 & 2|2|2|7 & 17|131 & 37|463 & 37463 \\ 2 & 57 & 3|19 & 11|29 & 1129 \\ 1 & 58 & 2|29 & 229 \\ 0 & 59 & 59 \\ 2 & 60 & 2|2|3|5 & 3|5|149 & 35149 \\ \hline \end{array}\]
Where \(m\) is the number of transforms needed for the new number to be prime.
It can be seen that some numbers lead to previous numbers and therefore arrive at the same prime number to terminate the sequence. The steps till the resulting number is prime has the sequence \(A(n)\)=0,0,0,2,0,1,0,13,2,4,0,1,0,5,4,4,0,1,0,15,1,1,0,2,3,4,4,1,0,2,0,2,1,16,3,2,0,2,1,9,0,2,0,9,6,1,0,0,-,2,1,1,0,1,2,3,2,1,0,2,... as the prime numbers require \(0\) transforms to be prime.
We could call this measure \(A(n)\) the anti-primality of \(n\), although, at the moment just because all prime numbers express anti-primality of \(0\). But does the index even mean anything like this...
Comparing the divisors sequence \(\sigma_0(n)\) and the \(A(n)\) sequence, we observe that:
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8,
0, 0, 0, 2, 0, 1, 0, 13,2, 4, 0, 1, 0, 5, 4, 4, 0, 1, 0, 15,1 ,1 ,0 ,2 ,3 ,4 ,4 ,1 ,0, 2,
2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12,
0, 2, 1, 16,3, 2, 0, 2, 1, 9, 0, 2, 0, 9, 6, 1, 0, 15, -, 2, 1, 1, 0, 1, 2, 3, 2, 1, 0, 2,
In the first \(40\) terms [not that many, this is speculative]
Every zero in \(A\) after the first is matched with a \(2\) in the divisor sequence, this is the primality of a number and is built in.
Every \(1\) in \(A\) has a corresponding divisor term of either \(4\) or \(6\). [true for first \(8\) \(1\)’s]
Every \(2\) in \(A\) has a corresponding divisor term of either \(3\), \(4\), \(6\), \(8\) or \(9\).
Every \(3\) has \(3\) or \(4\).
Every \(4\) has \(4\) or \(5\).
\(6\) has \(4\)
.. \(9\) has \(8\)
..
Find more terms...
Require a counter proof of the statement, “any non-prime whose ordered concatenation is prime has either \(4\) or \(6\) divisors”.
Counterproof: 531832651281459 has 128 divisors. But it’s prime factors are 3*3*3*11*139*653*3863*5107, which have an ordered concatenation of 3331113965338635107 which is prime.
Better counter proof, 54, has 8 divisors and has concatenation 2333 which is prime.
I like the region \(53\) to \(59\), as we have transforms till prime \(0,1,2,3,2,1,0\).