Puzzle 1043. Another puzzle about Keith numbers

Paolo Lava sent the following nice puzzle.

For the definition of Keith numbers have a look to A007629.

For n-th power analog of Keith numbers see A274769, A274770 and from A281915 to A281921.

Now, I search for the minimum prime p such that by applying the Keith Number Process (KNP) to p^n, with n>0, we reach p. Let us say, KNP(p^n) = p

"I take the definition from A007629:
Numbers n>9 with following property: form a sequence b(i) whose initial terms are the t digits of n, later terms given by rule that b(i) = sum of t previous terms; then n itself appears in the sequence.

E.g.
197 is a Keith number since sequence starts 1, 9, 7. Then 1+9+7=17. So 1,9,7,17. Again 9+7+17=33. So 1,9,7,17,33. Again 7+17+33=57, 17+33+57=107, 33+57+107=197 so 1,9,7,17,33,57,107,197 ... that is the number we started from.

I identify this process with the notation KNP (Keith Number Process)."

Here below the table I got for 1<n<100:

n	p
1	2
2	37
3	17
4	7
5	109
6
7	31
8	80051
9	71
10	97
11	107
12	13093
13	103
14	127
15	107
16	163
17	991
18	181
19	157
20	181
21	199
22	193
23	271
24	31663
25	211
26	307
27	307
28
29	673
30	8297
31	331
32	811
33	359
34	463
35
36	473741
37	421
38	2243
39	449
40	1031
41	11503
42	487
43	461
44	523
45	503
46	1171
47	1279
48	631
49	661
50	32323
51	321221
52	739
53	683
54	739
55	677
56	1571
57	719
58	1709
59	8237
60	185303
61
62	1759
63	827
64	829
65	422749
66
67	859
68	44687
69	9769
70	2099
71	991
72	108649
73	853
74	1009
75	2281
76	1093
77	1061
78	1117
79	1031
80	23687
81
82	1231
83	1151
84
85	1051
86	2591
87	1187
88	2791
89	1151
90
91
92	2803
93	3061
94	1303
95
96	14107
97	1237
98	3203
99	3061
100	1489

Q1. Is there any prime for n = 6, 28, 35, 61, 66, 81, 84, 90, 91, 95 ?
I guess there is always at least a p for any n but searching for it is a time consuming task for my PC.

Variants
KNP(p(n)) = p(n+1)
11, 13
29, 31
31, 37
97, 101
797, 809
22073, 22079
381287, 381289
E.g. 797 -> 809
7 + 9 + 7 = 23;
9 + 7 + 23 = 39;
7 + 23 + 39 = 69;
23 + 39 + 69 = 131;
39 + 69 + 131 = 239;
69 + 131 + 239 = 439;
131 + 239 + 439 = 809.
KNP(p(n)) = p(n-1)
11, 13
131, 137
6899, 6907
1963267, 1963277
KNP(p(n)+p(n+1)) = p(n+2)
2,3,5
12487, 12491, 12497
KNP(p(n)+p(n+1)) = p(n-1)
3,5,7
53,59,61
103,107,109
733, 739, 743
KNP(p(n)*p(n+1)) = p(n+2)
73,79,83
97,101,103
KNP(p(n)*p(n+1)) = p(n-1)
13,17,19
19,23,29
3433,3449,3457
KNP(p(n) U p(n+1)) = p(n+2)
2,3,5
19,23,29
163,167,173
311,313,317
30841,30851,30853
31237,31247,31249
3438973,3438091,3438103
KNP(p(n) U p(n+1)) = p(n-1)
107,109,113

Q2. Other values for these variants?