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 is prime, so this number, 110111111101, is an ænlic prime. If there were more than one prime in such a 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.

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.