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

1,1

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.

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.

Formula

Intersection of A000040 and A108038.

Example

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

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]

Crossrefs

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

Sequence in context: A350179 A229289 A087634 * A291691 A178576 A360147

Adjacent sequences: A360929 A360930 A360931 * A360933 A360934 A360935

Keyword

nonn

Author

Rigoberto Florez, Feb 25 2023

Status

approved