1 2 = 1 * 2 3 = 3 2 = 2 3 = 1 * 3 4 = 2 * 2 5 = 5 12 = 2^2 * 3 13 = 1 * 13 14 = 2 * 7 15 = 3 * 5 16 = 2^4 12720 = 2^4 * 3 * 5 * 53 12721 = 1 * 12721 12722 = 2 * 6361 12723 = 3 * 4241 12724 = 4 * 3181 12725 = 5^2 * 509 19440 = 2^4 * 3^5 * 5 19441 = 1 * 19441 19442 = 2 * 9721 19443 = 3 * 6481 19444 = 4 * 4861 19445 = 5 * 3889 19446 = 2 * 3 * 7 * 463 5516280 = 2^3 * 3^2 * 5 * 7 * 11 * 199 5516281 = 1 * 5516281 5516282 = 2 * 2758141 5516283 = 3 * 1838761 5516284 = 4 * 1379071 5516285 = 5 * 1103257 5516286 = 6 * 919381 5516287 = 7 * 788041 5516288 = 2^10 * 5387 7321991040 = 2^7 * 3^2 * 5 * 7 * 13 * 61 * 229 7321991041 = 1 * 7321991041 7321991042 = 2 * 3660995521 7321991043 = 3 * 2440663681 7321991044 = 4 * 1830497761 7321991045 = 5 * 1464398209 7321991046 = 6 * 1220331841 7321991047 = 7 * 1045998721 7321991048 = 8 * 915248881 7321991049 = 3^2 * 79 * 10298159 363500177040 = 2^4 * 3^3 * 5 * 7^3 * 11 * 13 * 47 * 73 363500177041 = 1 * 363500177041 363500177042 = 2 * 181750088521 363500177043 = 3 * 121166725681 363500177044 = 4 * 90875044261 363500177045 = 5 * 72700035409 363500177046 = 6 * 60583362841 363500177047 = 7 * 51928596721 363500177048 = 8 * 45437522131 363500177049 = 9 * 40388908561 363500177050 = 2 * 5^2 * 59 * 2203 * 55933 2394196081200 = 2^4 * 3^2 * 5^2 * 7 * 11 * 17 * 197 * 2579 2394196081201 = 1 * 2394196081201 2394196081202 = 2 * 1197098040601 2394196081203 = 3 * 798065360401 2394196081204 = 4 * 598549020301 2394196081205 = 5 * 478839216241 2394196081206 = 6 * 399032680201 2394196081207 = 7 * 342028011601 2394196081208 = 8 * 299274510151 2394196081209 = 9 * 266021786801 2394196081210 = 10 * 239419608121 2394196081211 = 11 * 29 * 41 * 79 * 2317171 3163427380990800 = 2^4 * 3^2 * 5^2 * 7^2 * 11 * 281 * 433 * 13399 3163427380990801 = 1 * 3163427380990801 3163427380990802 = 2 * 1581713690495401 3163427380990803 = 3 * 1054475793663601 3163427380990804 = 4 * 790856845247701 3163427380990805 = 5 * 632685476198161 3163427380990806 = 6 * 527237896831801 3163427380990807 = 7 * 451918197284401 3163427380990808 = 8 * 395428422623851 3163427380990809 = 9 * 351491931221201 3163427380990810 = 10 * 316342738099081 3163427380990811 = 11 * 287584307362801 3163427380990812 = 2^2 * 3 * 13 * 23 * 37 * 47 * 977 * 518933 n + i = i * p iCalculation with Words : Doric Columns of Primes
I propose that these striking, delightful columns of primes which are distinguished landmarks of rectilinearity in the beauty found in the complex mixture of regularity and irregularity of the stream of the factors of the integers should be known as Doric Columns of Primes, because of their shape and, especially, in honor of their creative architect, Charlie Dorian.
There were four most significant features of the search algorithm used. Firstly, elementary considerations were used to reduce the number of candidate solutions. For example, n[11] must be divisible by 55440 = 16 * 9 * 5 * 7 * 11. Secondly, and most importantly, for each candidate, as each potentially disqualifying factor was considered, the integers in the entire block were examined, with the most likely factors being considered first. This technique is called Gang Factoring. Generally, the smaller factors are more likely to be misplaced in the block. In seeking n[11], only one candidate in more than 400 required consideration of any factor larger than 29. Thirdly, the small factors were sieved using integer arithmetic, without multiplication or division operations, just 8bit subtractions and comparisons. Fourthly, highly optimized assembler code allowed the execution of only about 6 instructions to eliminate each small factor. The final algorithm, an unsuccessful search for n[12], was about 300 billion times more efficient than the one first implemented. As a result, running on a V20 and 8087 equipped 4.77 MHz IBMPC, the entire search, which extended to 2^53, took only a few months of execution time.
Open questions:
1) Do arbitrarily tall Doric Columns exist?
2) If not, how many and how large?

2000 Mathematics Subject Classification : Primary 11Y11 , 11A51 ; Secondary 11Y05 , 11A05 , 11A41 , 11N05 , 11N25 , 11Y11 , 11Y50 , 11Y55 .
This calculation was proposed and reported on the NUMBER Conference
of the MIX Bulletin Board System of the Capital PC User Group .
The illustrious Roger Fajman was the SYSOP of MIX , the Member
Information eXchange .
I was the moderator of the conference .
This calculation arose from a spirit of fun in part engendered by
Oystein Ore's Number Theory and Its History , McGrawHill , 1948 ;
Dover reprint , 1988 , ISBN: 9780486656205 .
Walter Nissen
written between 19910626 and 19910812
posted 20080616