Letting Mathematica sit and run overnight, I find the "weakly twin primes" 89(3|5)91959 with 8 digits, and 51951247(1|3), 5(3|6)1324041, 699023(7|8)91, and 87448(1|7)011 with 9 digits. That is, these pairs of numbers are each prime, they differ in one digit, and you cannot change either of them into any other prime by changing a digit. Searching OEIS for the first of these, 89391959, turns up http://oeis.org/A158576, "number of components of the graph P(n,10)", which is of course precisely the sequence Neil just asked for. We'd have found this yesterday, if the comments section for that sequence had also explicitly listed one of the weakly prime 6-digit numbers. And the sequence author is funster W. Edwin Clark, who is clearly not paying enough attention to our ramblings. (Yes, Neil, submitted as https://oeis.org/draft/A253269.) --Michael On Thu, Apr 30, 2015 at 8:26 PM, Andrew Trevorrow <andrew@trevorrow.com> wrote:

Michael Kleber wrote:

...

... It takes about 5 seconds to produce 294001, 505447, 584141, 604171.

Even the first term is enough to find http://oeis.org/A050249 ...

The full set of isolated 6-digit primes is 294001, 505447, 584141, 604171, 929573, 971767.

Note that A050249 doesn't include 929573 because 029573 isn't prime. See http://oeis.org/A158124 for the sequence of isolated primes where the 1st digit can't be changed to 0.

Many moons ago I wrote a small Mac app called Alchemy for finding word ladders. It lets you define the valid characters in a word, so by choosing 0..9 it's easy to create a "word" list containing prime numbers for finding prime ladders. The app also lets you find isolated primes, or search the shortest ladders for each pair of primes and find the longest examples:

no. of digits: 2 3 4 5 6 longest ladder: 4 7 9 11 13 example: 23 389 2441 88259 440497 43 383 5441 88289 410497 47 883 5741 88789 416497 37 863 5701 88799 416477 463 3701 82799 416473 461 3709 12799 416873 761 3109 12791 406873 9109 92791 406883 9199 92761 706883 99761 705883 99721 705833 905833 995833

Sadly, Alchemy is so old it only runs on Mac OS 10.6 or older, but I hope to modernize it one day. (More details can be seen at http://www.trevorrow.com/alchemy/. If you do have an old Mac and want to play with Alchemy then let me know and I'll send you my Primes word list.)

Andrew

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