## Thursday, May 30, 2024

### 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.

## Monday, May 27, 2024

### 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)

## Wednesday, May 22, 2024

### 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.