A339315 a(n) is the smallest number k such that k^2+1 divided by its largest prime factor is equal to F(2*n-1) for n > 0, or 0 if no such k exists, where F(n) is the Fibonacci sequence.

1, 3, 8, 34, 55, 144, 610, 233, 12166, 2584, 4181, 68260, 46368, 75025, 3917414, 464656, 1346269, 16349962

Offset

1,2

Comments

a(n) is the smallest number k such that A248516(k) = A001519(n) for n > 0, or 0 if no such k exists, where A001519(n) = F(2*n-1) (bisection of the Fibonacci sequence), with F(n) = A000045(n).

We observe that a(2 + 3m) = A001519(1 + 3m) = A000045(1 + 6m) for m = 2, 3, 4, 5. For n = 6, this property no longer works.

For k > 0, a(3k - 1) is odd, a(3k) and a(3k+1) are even.

We observe that a(n)^2 + 1 is the product of two prime Fibonacci numbers for n = 2, 3, 4, 6, 7.

The first 18 terms of the sequence are Fibonacci numbers, except a(9), a(12), a(15), a(16) and a(18).

The corresponding sequence b(n) = (a(n)^2+1)/ A001519(n) is 2, 5, 13, 89, 89, 233, 1597, 89, 92681, 1597, 1597, 162593, 28657, 28657, 29842993, 160373, 514229. We observe that a majority of terms of b(n) are prime Fibonacci numbers, except b(9), b(12), b(15) and b(16).

Links

Table of n, a(n) for n=1..18.

Example

a(4) = 34 because 34^2 + 1 = 13*89 = 1157, and 1157/89 = 13 = A248516(34) = A001519(4).
A curiosity: a(22) = 1134903170 = F(45) with F(45)^2 + 1 = F(43)*F(47) where F(43) and F(47) are prime Fibonacci numbers.

Maple

with(numtheory):with(combinat, fibonacci):
nn:=100:n0:=20:
for n from 1 to n0 do:
  ii:=0:
  for m from 1 to 10^10 while(ii=0) do:
   x:=m^2+1:y:=factorset(x):n1:=nops(y):
   z:=x/y[n1]:
    if z = fibonacci(2*n-1)
     then
     ii:=1:printf(`%d %d \n`, n, m):
     else
    fi:
  od:
od:

Prog

(PARI) a(n) = {my(k=1, f=fibonacci(2*n-1)); while ((k^2+1)/vecmax(factor(k^2+1)[, 1]) != f, k++); k; } \\ Michel Marcus, Nov 30 2020

Crossrefs

Cf. A000045, A001519, A001906, A002522, A005478, A014442, A245236, A245306, A248516, A281618.

Sequence in context: A184255 A001120 A302629 * A322303 A117722 A231856

Adjacent sequences: A339312 A339313 A339314 * A339316 A339317 A339318

Keyword

nonn,hard

Author

Michel Lagneau, Nov 30 2020

Status

approved