A386601 - OEIS

%I #36 Nov 06 2025 15:59:12

%S 2,3,5,7,19,31,53,97,131,227,353,449,661,797,883,919,1009,1033,1213,

%T 1307,2797,3221,3529,3877,4919,5099,5443,5477,5657,5741,6131,7211,

%U 7699,7879,7963,8353,9091,9433,9811,9887,10169,10729,11257,12451,12517,12659,14683

%N Primes that change into other primes of the same length at each of the four possible stages when all of their digits simultaneously shift along the ...8-9-0-1-2-3-4-5-6-7-8-9-0-1-2-3... cycle. No leading zeros and so no differing lengths of the changing primes are allowed.

%C Primes of length 1, 2, 3, 4, and 5 in this sequence fall into groups of size 1, 1, 2, 6, and 20 respectively. It appears that this number is growing by the growth of the digit count.

%e The single digit prime 2 changes through 3, 5, and 7, all of which mutually change into each other and 2, each being a prime. This is a full cycle of digit change with four primes as a result.

%e The 19, 31, 53, 97 two-digit primes also transform into each other when their digits simultaneously change along the 0-1-2-3-4-5-6-7-8-9-0-1-2-3... cycle.

%e The 131, 353, 797, 919 three-digit primes also form four transformations into each other by the rule:

%e .

%e ...9 - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 0 - 1 - 2 - 3...

%e            ^       ^

%e           1st     2nd

%e          digit   digit      131, a prime number

%e            ^                    <- Shift ->

%e           3rd

%e          digit

%e .

%e ...9 - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 0 - 1 - 2 - 3...

%e                    ^       ^

%e                   1st     2nd

%e                   digit   digit      353, a prime number

%e                    ^                     <- Shift ->

%e                   3rd

%e                  digit

%e .

%e ...9 - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 0 - 1 - 2 - 3...

%e                                    ^       ^

%e                                   1st     2nd

%e                                  digit   digit      797, a prime number

%e                                    ^                    <- Shift ->

%e                                   3rd

%e                                  digit

%e .

%e ...9 - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 0 - 1 - 2 - 3...

%e                                            ^       ^

%e                                           1st     2nd

%e                 919, a prime number      digit   digit

%e                     <- Shift ->            ^

%e                                           3rd

%e                                          digit

%e .

%e One of the six groups of the four-digit primes, as a sample:

%e 3877, 5099, 7211 and 9433.

%e .

%e One of the twenty groups of the five-digit primes, as a sample:

%e 20287, 42409, 64621, 86843.

%e .

%e Examples for failed leading zero candidates:

%e 2141, 4363, 8707, and 0929. Here, 929, although a prime, but not a four-digit one. The necessary endings to 1, 3, 7, and 9 is fulfilled but this group is disqualified because of that.

%e 2797, 4919, 6131, 8353, and 0575 are generated by the digit shift. Here, the effectively three-digit 575 ending with 5 is not a prime anyway, so this full group of four primes is not disqualified.

%Y Cf. A000040.

%K nonn,base

%O 1,1

%A _Tamas Sandor Nagy_, Oct 22 2025