Stable primes

Subsequent to my Stable pandigital numbers blog, I will now share my calculation of the first 2486689 stable primes. Reference A373117 to understand what is meant here by "stable".

      1 2

      2 3

      3 5

      4 7

      5 11

      6 101

      7 113

      8 131

      9 151

     10 157

     11 179

     12 181

     13 191

     14 311

     15 313

     16 317

     17 353

     18 373

     19 383

     20 419

     21 421

    ...

2486669 999986513

2486670 999986719

2486671 999986927

2486672 999987323

2486673 999988133

2486674 999988547

2486675 999989149

2486676 999989519

2486677 999989981

2486678 999990841

2486679 999993641

2486680 999994081

2486681 999994613

2486682 999995261

2486683 999995629

2486684 999996071

2486685 999997249

2486686 999997457

2486687 999998059

2486688 999998683

2486689 999998801

The entire list, which is complete to 10^9, is here. Since there are 50847534 primes less than one billion, the stable primes in this range are 4.89% of that total.

Stable pandigital numbers

The recent OEIS A373117 addition had me wondering how many of the 3265920 pandigitals were "stable" (or "balanced" in Éric Angelini's article). I found 135914 such:

     1  1023469875
     2  1023487695

     3  1023495876
     4  1023497658
     5  1023569748
     6  1023578496
     7  1023579468
     8  1023584976
     9  1023587649
    10  1023596478
    11  1023649857
    12  1023685497
    13  1023746895
    14  1023748596
    15  1023749568
    16  1023764958
    17  1023765849
    18  1023845796
    19  1023847659
    20  1023865479

    21  1024368975
   ...

135894  9876253014
135895  9876305421
135896  9876314250
135897  9876321540
135898  9876324051
135899  9876325104
135900  9876340251
135901  9876350214
135902  9876351024
135903  9876403512
135904  9876405132
135905  9876405213
135906  9876412503
135907  9876431052
135908  9876432105
135909  9876502413
135910  9876503142
135911  9876510432
135912  9876513024
135913  9876520314
135914  9876521043

The entire list is here. The vertical blades in the above are the seventh digit in the first 21 terms and the fourth digit in the final 21 terms. There are eighteen different ways in which ten-digit integers might be stable. Here are the possible digit-multipliers, in pandigitals the frequency of their occurrence, and the [smallest index, largest index]:

(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)

(-1, 1, 2, 3, 4, 5, 6, 7, 8, 9)

(-1, 0, 1, 2, 3, 4, 5, 6, 7, 8)

(-2, -1, 1, 2, 3, 4, 5, 6, 7, 8)

(-2, -1, 0, 1, 2, 3, 4, 5, 6, 7)

(-3, -2, -1, 1, 2, 3, 4, 5, 6, 7)

(-3, -2, -1, 0, 1, 2, 3, 4, 5, 6)        =>  2010  [60132, 135914]

(-4, -3, -2, -1, 1, 2, 3, 4, 5, 6)       => 14892  [11501, 135847]

(-4, -3, -2, -1, 0, 1, 2, 3, 4, 5)       => 37712  [ 4507, 135595]

(-5, -4, -3, -2, -1, 1, 2, 3, 4, 5)      => 36873  [ 1194, 134669]

(-5, -4, -3, -2, -1, 0, 1, 2, 3, 4)      => 31768  [  345, 128927]

(-6, -5, -4, -3, -2, -1, 1, 2, 3, 4)     => 11393  [   45, 110619]

(-6, -5, -4, -3, -2, -1, 0, 1, 2, 3)     =>  1266  [    1,  60133]

(-7, -6, -5, -4, -3, -2, -1, 1, 2, 3)

(-7, -6, -5, -4, -3, -2, -1, 0, 1, 2)

(-8, -7, -6, -5, -4, -3, -2, -1, 1, 2)

(-8, -7, -6, -5, -4, -3, -2, -1, 0, 1)

(-9, -8, -7, -6, -5, -4, -3, -2, -1, 1)

Clockwise analog clock primes

Éric Angelini and Michael Branicky have added a version 2 of clockwise analog clock primes. I was sufficiently motivated thereby to extend Michael's a(59) [= a(28) of the original] having 1325 digits into the beyond.