Problems & Puzzles: Puzzles

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?

During the week from June26 to July 2, contributions came from Emmanuel Vantieghem, Simon Cavegn, Giorgios  Kalogeropoulos.

***

Emmanuel wrote:

I have only two answers to question 1 :
n = 6  -> p = 36013476739 (found in  A281917)
n = 28 -> p = 5318989651.

***

Simon wrote:

Q1
n:81 p:200908021
n:84 p:25471
Searched up to p=4000000007

***

Giorgios wrote:

Q1.
n=6    -> p=36013476739 (this one was already known, see A281917)
n=28  -> p=5318989651
n=61  -> p=11659149703
n=81  -> p=200908021
n=84  -> p=25471

***

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