There are many examples of families that only contain
composite numbers (only count the numbers > base).

Example 1: base 10,
family 4{6}9 (formula: (14*10^(n+1)+7)/3)

49 = 7 * 7

469 = 7 * 67

4669 = 7 * 667

46669 = 7 * 6667

466669 = 7 * 66667

4666669 = 7 * 666667

Example 2: base 10,
family 28{0}7 (formula: 28*10^(n+1)+7)

287 = 7 * 41

2807 = 7 * 401

28007 = 7 * 4001

280007 = 7 * 40001

2800007 = 7 * 400001

28000007 = 7 * 4000001

Example 3: base 9,
family {1} (formula: (9^n-1)/8)

11 = 2 * 5

111 = 7 * 14

1111 = 22 * 45

11111 = 67 * 144

111111 = 222 * 445

1111111 = 667 * 1444

11111111 = 2222 * 4445

111111111 = 6667 * 14444

1111111111 = 22222 * 44445

11111111111 = 66667 * 144444

111111111111 = 222222 * 444445

1111111111111 = 666667 * 1444444

Example 4: base 9,
family 3{8} (formula: 4*9^n-1)

38 = 5 * 7

388 = 18 * 21

3888 = 58 * 61

38888 = 188 * 201

388888 = 588 * 601

3888888 = 1888 * 2001

38888888 = 5888 * 6001

388888888 = 18888 * 20001

3888888888 = 58888 * 60001

38888888888 = 188888 * 200001

388888888888 = 588888 * 600001

3888888888888 = 1888888 * 2000001

Example 5: base 8,
family 1{0}1 (formula: 8^(n+1)+1)

11 = 3 * 3

101 = 5 * 15

1001 = 11 * 71

10001 = 21 * 361

100001 = 41 * 1741

1000001 = 101 * 7701

10000001 = 201 * 37601

100000001 = 401 * 177401

1000000001 = 1001 * 777001

10000000001 = 2001 * 3776001

100000000001 = 4001 * 17774001

1000000000001 = 10001 * 77770001

Example 6: base 9,
family {8}5 (formula: 9^(n+1)-4)

85 = 7 * 12

885 = 87 * 102

8885 = 887 * 1002

88885 = 8887 * 10002

888885 = 88887 * 100002

8888885 = 888887 * 1000002

88888885 = 8888887 * 10000002

888888885 = 88888887 * 100000002

8888888885 = 888888887 * 1000000002

88888888885 = 8888888887 * 10000000002

888888888885 = 88888888887 * 100000000002

8888888888885 = 888888888887 * 1000000000002

Example 7: base 11,
family 2{5} (formula: (5*11^n-1)/2)

25 = 3 * 9

255 = 2 * 128

2555 = 3 * 919

25555 = 2 * 12828

255555 = 3 * 91919

2555555 = 2 * 1282828

25555555 = 3 * 9191919

255555555 = 2 * 128282828

2555555555 = 3 * 919191919

25555555555 = 2 * 12828282828

255555555555 = 3 * 91919191919

2555555555555 = 2 * 1282828282828

Example 8: base 12,
family {B}9B (formula: 12^(n+2)-25)

9B = 7 * 15

B9B = 11 * AB

BB9B = B7 * 105

BBB9B = 11 * B0AB

BBBB9B = BB7 * 1005

BBBBB9B = 11 * B0B0AB

BBBBBB9B = BBB7 * 10005

BBBBBBB9B = 11 * B0B0B0AB

BBBBBBBB9B = BBBB7 * 100005

BBBBBBBBB9B = 11 * B0B0B0B0AB

BBBBBBBBBB9B = BBBBB7 * 1000005

BBBBBBBBBBB9B = 11 * B0B0B0B0B0AB

Example 9: base 14,
family B{0}1 (formula: 11*14^(n+1)+1)

B1 = 5 * 23

B01 = 3 * 395

B001 = 5 * 22B3

B0001 = 3 * 39495

B00001 = 5 * 22B2B3

B000001 = 3 * 3949495

B0000001 = 5 * 22B2B2B3

B00000001 = 3 * 394949495

B000000001 = 5 * 22B2B2B2B3

B0000000001 = 3 * 39494949495

B00000000001 = 5 * 22B2B2B2B2B3

B000000000001 = 3 * 3949494949495

Example 10: base 13,
family 3{0}95 (formula: 3*13^(n+2)+122)

395 = 14 * 2B

3095 = 7 * 58A

30095 = 5 * 7A71

300095 = 7 * 5758A

3000095 = 14 * 23A92B

30000095 = 7 * 575758A

300000095 = 5 * 7A527A71

3000000095 = 7 * 57575758A

30000000095 = 14 * 23A923A92B

300000000095 = 7 * 5757575758A

3000000000095 = 5 * 7A527A527A71

30000000000095 = 7 * 575757575758A

Example 11: base 16,
family {4}D (formula: (4*16^(n+1)+131)/15)

4D = 7 * B

44D = 3 * 16F

444D = D * 541

4444D = 7 * 9C0B

44444D = 3 * 16C16F

444444D = D * 540541

4444444D = 7 * 9C09C0B

44444444D = 3 * 16C16C16F

444444444D = D * 540540541

4444444444D = 7 * 9C09C09C0B

44444444444D = 3 * 16C16C16C16F

444444444444D = D * 540540540541

Example 12: base 16,
family {C}D (formula: (4*16^(n+1)+1)/5)

CD = 5 * 29

CCD = 71 * 1D

CCCD = 1E1 * 6D

CCCCD = 18D * 841

CCCCCD = 64D * 2081

CCCCCCD = 7F01 * 19CD

CCCCCCCD = 1FE01 * 66CD

CCCCCCCCD = 198CD * 80401

CCCCCCCCCD = 664CD * 200801

CCCCCCCCCCD = 7FF001 * 199CCD

CCCCCCCCCCCD = 1FFE001 * 666CCD

CCCCCCCCCCCCD = 1998CCD * 8004001

Example 13: base 17,
family 1{9} (formula: (25*17^n-9)/16)

19 = 2 * D

199 = B * 27

1999 = 2 * D4D

19999 = AB * 287

199999 = 2 * D4D4D

1999999 = AAB * 2887

19999999 = 2 * D4D4D4D

199999999 = AAAB * 28887

1999999999 = 2 * D4D4D4D4D

19999999999 = AAAAB * 288887

199999999999 = 2 * D4D4D4D4D4D

1999999999999 = AAAAAB * 2888887

Example 14: base 19,
family 1{6} (formula: (4*19^n-1)/3)

16 = 5 * 5

166 = D * 1I

1666 = 5 * 515

16666 = CD * 1II

166666 = 5 * 51515

1666666 = CCD * 1III

16666666 = 5 * 5151515

166666666 = CCCD * 1IIII

1666666666 = 5 * 515151515

16666666666 = CCCCD * 1IIIII

166666666666 = 5 * 51515151515

1666666666666 = CCCCCD * 1IIIIII

Example 15: base 25,
family 2{1} (formula: (49*25^n-1)/24)

21 = 3 * H

211 = 14 * 1J

2111 = 2N * HC

21111 = 144 * 1IJ

211111 = 2MN * HCC

2111111 = 1444 * 1IIJ

21111111 = 2MMN * HCCC

211111111 = 14444 * 1IIIJ

2111111111 = 2MMMN * HCCCC

21111111111 = 144444 * 1IIIIJ

211111111111 = 2MMMMN * HCCCCC

2111111111111 = 1444444 * 1IIIIIJ

Example 16: base 36,
family O{Z} (formula: 25*36^n-1)

OZ = T * V

OZZ = 4Z * 51

OZZZ = TZ * U1

OZZZZ = 4ZZ * 501

OZZZZZ = TZZ * U01

OZZZZZZ = 4ZZZ * 5001

OZZZZZZZ = TZZZ * U001

OZZZZZZZZ = 4ZZZZ * 50001

OZZZZZZZZZ = TZZZZ * U0001

OZZZZZZZZZZ = 4ZZZZZ * 500001

OZZZZZZZZZZZ = TZZZZZ * U00001

OZZZZZZZZZZZZ = 4ZZZZZZ * 5000001

However, there are also
many families, which have no known prime (or PRP) > base, neither
can be proven that they only contain composite numbers (only count
the numbers > base), e.g.

base 11, family 5{7}

base 13, family 9{5}

base 13, family A{3}A

base 16, family {3}AF

base 16, family {4}DD

base 17, family 1{7}

base 17, family F1{9}

base 18, family C{0}C5

You can try to find a
prime (or PRP) > base in these families.