*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.

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