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A360932 Primes of the form H(m,k) = F(k+1)*F(m-k+2) - F(k)*F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.

%I A360932 #50 Aug 14 2025 17:46:19
%S A360932 2,3,5,7,11,13,23,29,37,41,47,89,97,103,107,157,167,173,199,233,443,
%T A360932 521,733,1597,1741,1867,1871,1877,2207,3037,3571,7841,7919,7951,9349,
%U A360932 11933,12823,28657,33503,50549,54277,54287,54293,54319,54497,55717,142099
%N A360932 Primes of the form H(m,k) = F(k+1)*F(m-k+2) - F(k)*F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.
%C A360932 This sequence appears in the triangle A108038 in this order (reading by rows): 3, 2, 7, 5, 11, 13, 29, 23, 47, 37, 41, 97, 107, 103, 89, 199, 157, 173, 167.
%C A360932 Are there infinitely many primes of the form H(m,k)?
%C A360932 This sequence appears within the determinant Hosoya triangle.
%H A360932 Robert Israel, <a href="/A360932/b360932.txt">Table of n, a(n) for n = 1..9313</a>
%H A360932 Hsin-Yun Ching, Rigoberto Florez, F. Luca, Antara Mukherjee, and J. C. Saunders, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/60-5/ching.pdf">Primes and composites in the determinant Hosoya triangle</a>, Fibonacci Quarterly, 2023.
%F A360932 Intersection of A000040 and A108038.
%e A360932 29 is a term because it is prime and A108038(8,2) = H(8,2) = 29.  Also A108038(8,7) = H(8,7) = 29.
%p A360932 Res:= {}:
%p A360932 M:= 50: # for terms <= F(M)
%p A360932 fmax:= combinat:-fibonacci(M):
%p A360932 T[1]:= [1,1]:
%p A360932 T[2]:= [1,3,1]:
%p A360932 for i from 3 to M do
%p A360932   t1:= [op(T[i-1][1..i-1] + T[i-2][1..i-1]),T[i-1][i],0];
%p A360932   t2:= ListTools:-Reverse(t1);
%p A360932   T[i]:= zip(max,t1,t2);
%p A360932   Res:= Res union convert(select(t -> t <= fmax and isprime(t), T[i][1..ceil((i+1)/2)]),set)
%p A360932 od:
%p A360932 sort(convert(Res,list)); # _Robert Israel_, Mar 14 2024
%t A360932 H[r_, k_] := Det[{{Fibonacci[r-k+2], Fibonacci[r-k+1]}, {Fibonacci[k], Fibonacci[k+1]}}];  DeterminantPrimes[t_, m_] :=  Table[If[PrimeQ[H[r,k]],H[r,k], Unevaluated[Sequence[]]], {r,t,m}, {k,1,Ceiling[r/2]}]; ListOfPrimes[t_,m_]:=  Sort[DeleteDuplicates[Flatten[DeterminantPrimes[t, m]]]]; ListOfPrimes[2, 100]
%t A360932 Select[Union[Flatten[Table[Fibonacci[k+1]Fibonacci[m-k+2]-Fibonacci[k]Fibonacci[m-k+1],{m,0,40},{k,0,m}]]],PrimeQ] (* _Harvey P. Dale_, Aug 14 2025 *)
%Y A360932 Cf. A000040, A000045, A005478 (subsequence), A108038, A153892, A067331.
%K A360932 nonn
%O A360932 1,1
%A A360932 _Rigoberto Florez_, Feb 25 2023