The algorithm that I got Gemini to create and that I used in the post 23 Revisited was the result of the following prompt:
Write a program in SageMath that determines all the composite, positive integers in a given range that contain the digits 2 and 3 in ascending order from left to right (but not necessarily consecutively) and whose prime factors also all contain the digits 2 and 3 in ascending order but not necessarily consecutively. The default range can be from 1 to 40000. The output should a table showing number and factors followed by a comma-separated list of the qualifying numbers.
Instead of the prime 23, other prime numbers can be investigated. Composite numbers could be investigated too but this would lead to rather large numbers and so I'll restrict this investigation to prime numbers but I'll increase the range from 1 to one million.
Here are the results for 11 (permalink):
Number | Prime Factors------------------------------110231 | 11, 911111221 | 11, 10111112321 | 11, 10211117821 | 11, 10711128711 | 11, 11701129811 | 11, 11801131153 | 11, 11923131197 | 11, 11927171121 | 211, 811211321 | 11, 19211231121 | 11, 21011232111 | 11, 21101261143 | 113, 2311351119 | 311, 1129451121 | 11, 41011511621 | 11, 46511781121 | 11, 71011892111 | 11, 81101
Here are the results for 13 (permalink):
Here are the results for 17 (permalink):Number | Prime Factors------------------------------13169 | 13, 101316913 | 13, 130122139 | 13, 131113269 | 13, 8713131729 | 13, 10133131807 | 13, 10139132703 | 131, 1013134069 | 13, 10313136669 | 13, 10513137069 | 113, 1213137969 | 13, 10613138781 | 137, 1013147013 | 113, 1301161369 | 13, 12413169013 | 13, 13001171301 | 13, 13177171379 | 13, 13183172133 | 13, 13241180713 | 13, 13901190013 | 139, 1367211303 | 131, 1613313469 | 13, 24113313729 | 13, 24133313781 | 13, 24137360139 | 13, 2131407213 | 313, 1301413473 | 313, 1321421369 | 13, 32413661369 | 313, 2113797513 | 613, 1301813397 | 13, 4813813451 | 613, 1327
Number | Prime Factors------------------------------51731 | 17, 179117589 | 17, 6917173009 | 17, 10177178313 | 17, 617198917 | 17, 11701217843 | 179, 1217291703 | 17, 17159295817 | 17, 17401301733 | 17, 17749308417 | 179, 1723351713 | 17, 1217368917 | 17, 21701506617 | 17, 1753517021 | 17, 1789541753 | 317, 1709717043 | 17, 42179
Number | Prime Factors------------------------------19361 | 19, 101936119 | 19, 1901119467 | 193, 619121943 | 197, 619193667 | 19, 10193194629 | 191, 1019196667 | 193, 1019251929 | 191, 1319361019 | 19, 19001361969 | 19, 19051366719 | 19, 19301370519 | 19, 19501371963 | 19, 19577376219 | 19, 19801381947 | 193, 1979519329 | 191, 2719601967 | 193, 3119714419 | 19, 1979719473 | 19, 1993752419 | 19, 199796519 | 419, 1901
I won't continue with primes larger than 23 because really what I want is an algorithm that will identify a sequence of digits that form a prime within a number and then check that the same digit sequence occurs with each of its prime factors. The number being checked is thus composite. Here is a permalink to the program that I've now included in my daily number analysis.
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