[ November 29, 2014 ]
OEIS Reference Table for primes of the form 10^n –/+ d
Searching for some patterns

A002283	10^n – 1	1	10^n + 1	¬
A093172	10^n – 3	3	10^n + 3	A159352
¬	10^n – 7	7	10^n + 7	A159031
A093177	10^n – 9	9	10^n + 9	¬
¬	10^n – 11	11	10^n + 11	¬
¬	10^n – 13	13	10^n + 13	¬
¬	10^n – 17	17	10^n + 17	¬
¬	10^n – 19	19	10^n + 19	¬
¬	10^n – 21	21	10^n + 21	A138861
¬	10^n – 23	23	10^n + 23	¬
A176987	10^n – 27	27	10^n + 27	¬
¬	10^n – 29	29	10^n + 29	¬
¬	10^n – 31	31	10^n + 31	¬
¬	10^n – 33	33	10^n + 33	¬
¬	10^n – 37	37	10^n + 37	¬
¬	10^n – 39	39	10^n + 39	¬
¬	10^n – 41	41	10^n + 41	¬
¬	10^n – 43	43	10^n + 43	¬
¬	10^n – 47	47	10^n + 47	¬
¬	10^n – 49	49	10^n + 49	¬
¬	10^n – 51	51	10^n + 51	¬
¬	10^n – 53	53	10^n + 53	¬
A177417	10^n – 57	57	10^n + 57	¬
A177418	10^n – 59	59	10^n + 59	¬
¬	10^n – 61	61	10^n + 61	¬
¬	10^n – 63	63	10^n + 63	¬
¬	10^n – 67	67	10^n + 67	¬
¬	10^n – 69	69	10^n + 69	¬
¬	10^n – 71	71	10^n + 71	¬
¬	10^n – 73	73	10^n + 73	¬
¬	10^n – 77	77	10^n + 77	¬
¬	10^n – 79	79	10^n + 79	¬
¬	10^n – 81	81	10^n + 81	¬
¬	10^n – 83	83	10^n + 83	¬
¬	10^n – 87	87	10^n + 87	¬
¬	10^n – 89	89	10^n + 89	¬
¬	10^n – 91	91	10^n + 91	¬
¬	10^n – 93	93	10^n + 93	¬
¬	10^n – 97	97	10^n + 97	¬
¬	10^n – 99	99	10^n + 99	¬

Detecting patterns

The midcolumn reveals immediately some patterns.
When there exist more than one prime for 10^m – d
and more than one prime for 10^n + d then the displacements d
are all congruent to 3 mod 6 (or to 0 mod 3).
The sequence (highlighted with cells in light green background color) looks like this :
3, 9, 21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99
{ After the initial 3 only odd composites divisible by 3 appear. }
A second pattern pops up when jumping from one to the next term :
+6, +12, +6, +6, +6, +12, +6, +6, +6, +12, +6, +6, +6
Is this [+6, +12, +6, +6] an infinite pattern ?

When there exist more than one prime for 10^m – d
but not for 10^n + d then the displacements d
are all congruent to 5 mod 6.
The sequence (aligned leftwards in the midcolumn of the table) looks like this :
11, 17, 23, 29, 41, 47, 53, 59, 71, 77, 83, 89
{ only one composite in the list... }
An analogue second pattern shows up :
+6, +6, +6, +12, +6, +6, +6, +12, +6, +6, +6
Same question : is this [+6, +6, +6, +12] an infinite pattern ?

When there exist more than one prime for 10^n + d
but not for 10^m – d then the displacements d
are all congruent to 1 mod 6.
The sequence (aligned rightwards in the midcolumn of the table) looks like this :
1, 7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 97
{ Equivalent to A004611 - Divisible only by primes congruent to 1 mod 3. }
An analogue second pattern shows up :
+6, +6, +6, +12, +6, +6, +6, +12, +6, +6, +6, +12, +6
Same question : is this [+6, +6, +6, +12] an infinite pattern ?

ps. displacements that are multiples of 5 are not considered.
These can never give rise to more than one prime, namely 5 itself, in either forms.

If there is more to say about this topic please write me and I'll add your comments.