Counts for 5-digit primes
Vertical: digit sum (multiples of 3 omitted)
Horizontal: first digit

    |   1   2   3   4   5   6   7   8   9 | total
----+-------------------------------------+------
  2 |                                     |    0
  4 |   1   3                             |    4
  5 |   7   4   1                         |   12
  7 |  11  10   3   2   2                 |   28
  8 |  19  12   5   4   2   2   1         |   45
 10 |  26  24  15  12   9   3   4   1   1 |   95
 11 |  46  29  25  15  16   6   2   3   1 |  143
 13 |  61  48  43  30  20  15  10   6   3 |  236
 14 |  66  51  43  33  23  25  12  13   6 |  272
 16 |  85  70  61  46  44  38  25  24  18 |  411
 17 |  90  86  77  62  51  39  34  25  15 |  479
 19 | 106  93  88  73  72  58  54  44  42 |  630
 20 |  97 103  84  80  74  68  62  53  43 |  664
 22 |  90  83  94  97  79  76  84  76  63 |  742
 23 |  89  93 101  88  88  80  77  73  68 |  757
 25 |  75  78  82  84  91  85  88  85  73 |  741
 26 |  61  67  65  79  94  87  87  83  83 |  706
 28 |  37  50  52  63  65  68  82  82  81 |  580
 29 |  32  32  49  54  64  66  72  74  85 |  528
 31 |  18  19  26  41  52  46  52  61  64 |  379
 32 |   9  17  19  36  34  46  57  58  65 |  341
 34 |   3   7  13  15  17  29  34  34  53 |  205
 35 |   4   3   9  11  12  23  27  35  42 |  166
 37 |       1   2   3   7   9  16  15  31 |   84
 38 |           1   1   7   6  11  16  20 |   62
 40 |               1       2   5  10  16 |   34
 41 |                   1   1   5   3   3 |   13
 43 |                           1   2   1 |    4
 44 |                                   2 |    2
----+-------------------------------------+------
tot |1033 983 958 930 924 878 902 876 879 | 8363

Max count is 757 for digit sum 23