Ænlic primes

Last month I showcased a very large term in OEIS sequence A039986. I didn't then have a special name for these numbers but I've given it some thought and finally settled on ænlic primes. Say what? A program-generated pronunciation [loud] for the Old English expression suggests (to my ear) a tad more ah than eh of that initial ae diphthong. The ending is like the word each. But I digress.

To recap, a term in A039986 is a prime number where no other permutation of its digits is prime — including permutations with leading zeros. For example, using two zero digits and ten one digits we can create the following 66 distinct permutations:

 1    1111111111 = 11*41*271*9091
 2   10111111111 = 101*421*237791
 3   11011111111 = 7*11*13*11000111
 4   11101111111 = 211*52611901
 5   11110111111 = 11*31*43*191*3967
 6   11111011111 = 41*101*271*9901
 7   11111101111 = 11*17*59417653
 8   11111110111 = 19*53*1667*6619
 9   11111111011 = 7*11*13^2*29*29443
10   11111111101 = 23*101*1931*2477
11   11111111110 = 2*5*11*41*271*9091
12  100111111111 = 11*31*293580971
13  101011111111 = 19*101*52637369
14  101101111111 = 7*11*13*17*281*21143
15  101110111111 = 41*269*271*33829
16  101111011111 = 11^2*2791*299401
17  101111101111 = 101*9901*101111
18  101111110111 = 11*59*155795239
19  101111111011 = 89*823*859*1607
20  101111111101 = 7*11*13*41*271*9091
21  101111111110 = 2*5*101*421*237791
22  110011111111 = 11*10181*982321
23  110101111111 = 101*1090110011
24  110110111111 = 7*11*13^2*293*28879
25  110111011111 = 97*659*1722557
26  110111101111 = 11*10010100101
27  110111110111 = 101*149*739*9901
28  110111111011 = 11*109*6047*15187
29  110111111101 = 110111111101
30  110111111110 = 2*5*7*11*13*11000111
31  111001111111 = 11*53*1097*173561
32  111010111111 = 101*181*6072431
33  111011011111 = 7^2*11*13*15842873
34  111011101111 = 383*289846217
35  111011110111 = 11^2*47*19520153
36  111011111011 = 31*101*3581*9901
37  111011111101 = 11*9013*1119707
38  111011111110 = 2*5*211*52611901
39  111100111111 = 11*19*531579479
40  111101011111 = 61*101*18032951
41  111101101111 = 7*11*13*23*71*67967
42  111101110111 = 1373*80918507
43  111101111011 = 11*10100101001
44  111101111101 = 101*241*461*9901
45  111101111110 = 2*5*11*31*43*191*3967
46  111110011111 = 11*41*271*909091
47  111110101111 = 83*101*2887*4591
48  111110110111 = 7*11*13*683*162517
49  111110111011 = 17*71*2777*33149
50  111110111101 = 11^3*83478671
51  111110111110 = 2*5*41*101*271*9901
52  111111001111 = 11*131*77106871
53  111111010111 = 101*5647*194813
54  111111011011 = 7*11*13*199*557789
55  111111011101 = 41*251*271*39841
56  111111011110 = 2*5*11*17*59417653
57  111111100111 = 11*10101009101
58  111111101011 = 101*1100109911
59  111111101101 = 7*11*13*111000101
60  111111101110 = 2*5*19*53*1667*6619
61  111111110011 = 11*1499*6738499
62  111111110101 = 17^2*101*3806609
63  111111110110 = 2*5*7*11*13^2*29*29443
64  111111111001 = 11*307*32902313
65  111111111010 = 2*5*23*101*1931*2477
66  111111111100 = 2^2*5^2*11*41*271*9091

Only #29 (110111111101) is prime and because this number contains all twelve digits (i.e., both zeros) it is an ænlic prime. [There is, for example, only one prime in the 21 permutations of two zeros and five ones but that prime (101111) is missing a zero so it is not ænlic.] If there were more than one prime in any such permutation list, they would not be ænlic. Using PARI code on the OEIS A039986 webpage, I generated 1141 ænlic primes less than 10^30. The program spent two weeks looking for 31-digit solutions — only to abort with a memory issue!

At that point I decided to write my own Mathematica program. Not a foolproof discovery engine that calculated all terms for a given digit-length — because such searches would take too long — but rather, a brute force effort that started with a given number of repdigits (1-9) and replaced up to two of the digits with something else (0-9, but not the repdigit). I can't be certain that this finds all of the ænlic primes that exist for the digit-length under consideration — but I think it probable.

The program is currently up to 10^350. I've preceded the decimal numbers with compact notation (where warranted: all but 44) and indexed the lot. I should be able to reach 10^500 but it will take another couple of weeks and will further deprive my "long compute" of the four cores on my fastest machine. [August 19: done!]

The number of ænlic primes of a given digit-length (4, 13, 34, 45, 68, 67, 47, 36, 40, 46, 33, 45, 35, 38, 32, 39, ...) is remarkably stable:

In my range — except for four near-the-start values — they are always greater than 20 and less than 48. I wonder if there exists a long-term trend.

More ænlic primes