Prime Hyperfibonacci numbers

[Jonathan Post]

A generalization of prime Fibonacci numbers (A005478) is the prime
hyperfibonacci numbers (primes in A136431).

Referring to the array a(k,n) =  Apply partial sum operator k times to
Fibonacci numbers, we see that every prime occurs in the n=2 column
(as it contains every positive integer).  So excluding n=2 and k=0
(A005478) we have the nontrivially prime hyperfibonacci numbers which
are not Fibonacci numbers.  These include:

k=1: primes in A000071 = {A000071(4) = 7}, no more through n = 36.

k=2: primes in A001924 = {A001924(3) = 7, A001924(7) = 79, A001924(25) = 514201}

k=3: primes in A014162 = {A014162(3) = 11, A014162(6) = 97,
A014162(16) = 17519}, no more through n = 30.

k=4: primes in A014166 = {A014166(4) = 41, A014166(13) = 10093,
A014166(14) = 16703}

k=5: primes in A053739 = {A053739(7) = 709, A053739(10) = 8273,
A053739(11) = 14323}, no more through n = 27.

k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683,
23945893(24) = 23945893}, no more through n = 27.

k=7: primes in A053296 = {A053296(3) = 37, A053296(6) = 967,
A053296(7) = 2267, A053296(12) = 127921, A053296(13) = 226007}, no
more through n = 27.

k=8: primes in A053308, none through n = 27.

k=9: primes in A053309, none through n = 26.

So, without going past k=9 or those values already in the existing
OEIS sequences other than A000045 (A000071, A001924, A014162, A014166,
A053739, A053295, A053296, A053308, A053309, A123736) 20 of the the
nontrivial hyperfibonacci primes include:

7, 11, 29, 37, 41, 79, 97, 709, 967, 2267, 2683, 8273, 10093, 14323,
16703, 17519, 127921, 226007, 514201, 23945893

Does someone want to code iterated application of the partial sum
operator k times to Fibonacci numbers, and test for primes?  I'd be
pleased to be co-author with whomsoever wants to do that coding and
primality testing.

Best,

Jonathan Vos Post