two integrals and a question about prime trinomials

[David Wilson]

Regarding A002624, it appears that:

For each prime p, there is a set of base-p digits D for such that p divides
a(n) iff the base-p representation of n includes one of the digits in D.
Apparently d in D implies p-1-d in D.

I don't have time to investigate, but I am curious as to whether a similar
phenomenon exists for prime powers.

The following table gives p and the associated D for p <= 1000.  For some
p, D is empty, indicating that no a(n) is divisible by p; these p are 
omitted
from the table.  I do not immediately see a pattern.

Assuming the above observations are correct the p column gives the primes
dividing some element of A002624.  This column does not match any existing
OEIS sequence.

       p  D

       3  {2}
       7  {3}
      17  {5 11}
      19  {4 14}
      41  {8 16 24 32}
      43  {9 33}
      47  {6 23 40}
      73  {9 13 18 54 59 63}
     107  {26 80}
     109  {47 61}
     113  {12 100}
     131  {7 123}
     173  {28 144}
     179  {44 134}
     191  {31 95 159}
     193  {51 82 110 141}
     199  {61 137}
     233  {27 46 77 155 186 205}
     269  {89 179}
     277  {90 186}
     281  {135 145}
     283  {98 184}
     293  {48 244}
     307  {122 184}
     311  {120 190}
     347  {121 225}
     373  {112 260}
     383  {91 191 291}
     401  {55 60 340 345}
     409  {63 345}
     419  {133 285}
     421  {48 156 264 372}
     439  {126 219 312}
     443  {179 263}
     457  {78 378}
     467  {184 282}
     503  {11 59 83 419 443 491}
     509  {84 424}
     521  {145 375}
     547  {195 351}
     563  {34 197 365 528}
     569  {14 554}
     593  {142 450}
     613  {44 568}
     617  {102 514}
     631  {195 435}
     653  {12 640}
     673  {96 576}
     691  {69 621}
     701  {340 360}
     709  {65 85 623 643}
     719  {30 123 595 688}
     739  {192 546}
     743  {160 296 446 582}
     809  {193 615}
     821  {118 702}
     823  {315 507}
     853  {49 803}
     857  {225 631}
     877  {273 603}
     881  {293 587}
     883  {317 565}
     907  {200 706}
     919  {208 710}
     929  {69 859}
     937  {100 836}
     941  {204 736}
     947  {94 269 349 456 490 597 677 852}
     953  {317 635}
     967  {36 930}



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