6-P-6 Primes And Beyond

What I mean by a 6-P-6 prime is a prime, greater than 2,  whose two adjacent composite numbers contain exactly six prime factors with multiplicity. There are only 30 such primes in the range up to 40000 and they are (permalink):

1889, 3079, 4591, 5023, 7649, 12689, 13751, 18089, 19249, 19889, 22193, 22639, 23057, 23311, 23561, 26839, 27919, 28027, 28751, 30449, 30941, 31121, 32993, 33641, 33967, 36251, 38177, 38431, 39799, 39929

Here are the details:

  previous                prime   next

  2^5 * 59                1889    2 * 3^3 * 5 * 7
  2 * 3^4 * 19            3079    2^3 * 5 * 7 * 11
  2 * 3^3 * 5 * 17        4591    2^4 * 7 * 41
  2 * 3^4 * 31            5023    2^5 * 157
  2^5 * 239               7649    2 * 3^2 * 5^2 * 17
  2^4 * 13 * 61           12689   2 * 3^3 * 5 * 47
  2 * 5^4 * 11            13751   2^3 * 3^2 * 191
  2^3 * 7 * 17 * 19       18089   2 * 3^3 * 5 * 67
  2^4 * 3 * 401           19249   2 * 5^3 * 7 * 11
  2^4 * 11 * 113          19889   2 * 3^2 * 5 * 13 * 17
  2^4 * 19 * 73           22193   2 * 3^4 * 137
  2 * 3 * 7^3 * 11        22639   2^4 * 5 * 283
  2^4 * 11 * 131          23057   2 * 3^3 * 7 * 61
  2 * 3^2 * 5 * 7 * 37    23311   2^4 * 31 * 47
  2^3 * 5 * 19 * 31       23561   2 * 3^2 * 7 * 11 * 17
  2 * 3^3 * 7 * 71        26839   2^3 * 5 * 11 * 61
  2 * 3^3 * 11 * 47       27919   2^4 * 5 * 349
  2 * 3^4 * 173           28027   2^2 * 7^2 * 11 * 13
  2 * 5^4 * 23            28751   2^4 * 3 * 599
  2^4 * 11 * 173          30449   2 * 3 * 5^2 * 7 * 29
  2^2 * 5 * 7 * 13 * 17   30941   2 * 3^4 * 191
  2^4 * 5 * 389           31121   2 * 3^2 * 7 * 13 * 19
  2^5 * 1031              32993   2 * 3^3 * 13 * 47
  2^3 * 5 * 29^2          33641   2 * 3^3 * 7 * 89
  2 * 3^3 * 17 * 37       33967   2^4 * 11 * 193
  2 * 5^4 * 29            36251   2^2 * 3^2 * 19 * 53
  2^5 * 1193              38177   2 * 3^3 * 7 * 101
  2 * 3^2 * 5 * 7 * 61    38431   2^5 * 1201
  2 * 3^3 * 11 * 67       39799   2^3 * 5^2 * 199
  2^3 * 7 * 23 * 31       39929   2 * 3 * 5 * 11^3

The algorithm linked to above is easily modified to find 7-P-7 primes that are surrounded by two composite numbers with exactly seven prime factors with multiplicity. There are only four in the range up to 40000 are these are 10529, 15391, 32561 and 35153. The details are (permalink):

  previous            prime   next

  2^5 * 7 * 47        10529   2 * 3^4 * 5 * 13
  2 * 3^4 * 5 * 19    15391   2^5 * 13 * 37
  2^4 * 5 * 11 * 37   32561   2 * 3^5 * 67
  2^4 * 13^3          35153   2 * 3^4 * 7 * 31

There are no 8-P-8 primes in the range up to 40000 but if we extend the range to 100000 we find one (permalink):

  previous         prime   next

  2^6 * 11 * 107   75329   2 * 3^5 * 5 * 31

These number properties of certain primes are not base-dependent. Obviously as the numbers get bigger there will be instances of 9-P-9 primes and beyond.