A386601 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.

2, 3, 5, 7, 19, 31, 53, 97, 131, 227, 353, 449, 661, 797, 883, 919, 1009, 1033, 1213, 1307, 2797, 3221, 3529, 3877, 4919, 5099, 5443, 5477, 5657, 5741, 6131, 7211, 7699, 7879, 7963, 8353, 9091, 9433, 9811, 9887, 10169, 10729, 11257, 12451, 12517, 12659, 14683

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..47.

Example

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.
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.
The 131, 353, 797, 919 three-digit primes also form four transformations into each other by the rule:
.
...9 - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 0 - 1 - 2 - 3...
           ^       ^
          1st     2nd
         digit   digit      131, a prime number
           ^                    <- Shift ->
          3rd
         digit
.
...9 - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 0 - 1 - 2 - 3...
                   ^       ^
                  1st     2nd
                  digit   digit      353, a prime number
                   ^                     <- Shift ->
                  3rd
                 digit
.
...9 - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 0 - 1 - 2 - 3...
                                   ^       ^
                                  1st     2nd
                                 digit   digit      797, a prime number
                                   ^                    <- Shift ->
                                  3rd
                                 digit
.
...9 - 0 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 0 - 1 - 2 - 3...
                                           ^       ^
                                          1st     2nd
                919, a prime number      digit   digit
                    <- Shift ->            ^
                                          3rd
                                         digit
.
One of the six groups of the four-digit primes, as a sample:
3877, 5099, 7211 and 9433.
.
One of the twenty groups of the five-digit primes, as a sample:
20287, 42409, 64621, 86843.
.
Examples for failed leading zero candidates:
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.
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.

Crossrefs

Cf. A000040.

Sequence in context: A087562 A375095 A154525 * A288717 A144561 A215156

Adjacent sequences: A386598 A386599 A386600 * A386602 A386603 A386604

Keyword

nonn,base

Author

Tamas Sandor Nagy, Oct 22 2025

Status

approved