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.

2, 3, 5, 7, 11, 13, 23, 29, 37, 41, 47, 89, 97, 103, 107, 157, 167, 173, 199, 233, 443, 521, 733, 1597, 1741, 1867, 1871, 1877, 2207, 3037, 3571, 7841, 7919, 7951, 9349, 11933, 12823, 28657, 33503, 50549, 54277, 54287, 54293, 54319, 54497, 55717, 142099

Offset: 1

Author

Rigoberto Florez, Feb 25 2023

Keywords

• nonn

Comments

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.

Are there infinitely many primes of the form H(m,k)?

This sequence appears within the determinant Hosoya triangle.

Examples

			29 is a term because it is prime and A108038(8,2) = H(8,2) = 29.  Also A108038(8,7) = H(8,7) = 29.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..9313
• Hsin-Yun Ching, Rigoberto Florez, F. Luca, Antara Mukherjee, and J. C. Saunders, Primes and composites in the determinant Hosoya triangle, Fibonacci Quarterly, 2023.

Crossrefs

Cf. A000040, A000045, A005478 (subsequence), A108038, A153892, A067331.

Programs

Maple

Res:= {}:
M:= 50: # for terms <= F(M)
fmax:= combinat:-fibonacci(M):
T[1]:= [1,1]:
T[2]:= [1,3,1]:
for i from 3 to M do
  t1:= [op(T[i-1][1..i-1] + T[i-2][1..i-1]),T[i-1][i],0];
  t2:= ListTools:-Reverse(t1);
  T[i]:= zip(max,t1,t2);
  Res:= Res union convert(select(t -> t <= fmax and isprime(t), T[i][1..ceil((i+1)/2)]),set)
od:
sort(convert(Res,list)); # Robert Israel, Mar 14 2024

Mathematica

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]
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 *)

Formula

Intersection of A000040 and A108038.