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A360932
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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.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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29 is a term because it is prime and A108038(8,2) = H(8,2) = 29. Also A108038(8,7) = H(8,7) = 29.
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MAPLE
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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:
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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