A281618 -id:A281618 - cp's OEIS frontend

A352290 Numbers m such that the greatest prime factor of m^2 + 1 is a Fibonacci number.

1, 2, 3, 5, 7, 8, 18, 34, 55, 57, 89, 123, 144, 233, 239, 322, 377, 411, 500, 568, 610, 746, 788, 843, 987, 1487, 1542, 1568, 1636, 2207, 2584, 2707, 3173, 3639, 3793, 3804, 3817, 4050, 4181, 4217, 4594, 4662, 5270, 5778, 6107, 6613, 8595, 8972, 10341, 10569

Offset: 1

Michel Lagneau, Mar 11 2022

nonn

A281618 is a subsequence.
The corresponding greatest prime Fibonacci factors of the sequence are 2, 5, 5, 13, 5, 13, 13, 89, 89, 13, 233, 89, 233, ...
The Fibonacci numbers of the sequence are 1, 2, 3, 5, 8, 34, 55, 89, 144, 233, 377, 610, 987, 2584, 4181, 10946, 17711, ... (subsequence of A000045).
The Lucas numbers of the sequence are 1, 2, 3, 7, 18, 123, 322, 843, 2207, 5778, 39603, 103682, ... (subsequence of A000032).
The prime numbers of the sequence are 2, 3, 5, 7, 89, 233, 239, 1487, 2207, 2707, 3793, 4217, 11789, 11981, 13763, ... including the prime Fibonacci numbers 2, 3, 5, 89, 233, 1066340417491710595814572169, ... (subsequence of A005478).

			18 is in the sequence because 18^2 + 1 = 5^2*13 and 13 is a Fibonacci number.

Cf. A000032, A000045, A002522, A005478, A014442, A248516, A338762, A339461.

q:= n-> (t-> ormap(issqr, [t+4, t-4]))(5*max(numtheory[factorset](n^2+1))^2):
select(q, [$1..12000])[];  # Alois P. Heinz, Mar 11 2022

With[{f = Fibonacci[Range[21]], m = f[[-1]]}, Select[Range[m], MemberQ[f, FactorInteger[#^2 + 1][[-1, 1]]] &]] (* Amiram Eldar, Mar 11 2022 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(m) = isfib(vecmax(factor(m^2+1)[,1])); \\ Michel Marcus, Mar 11 2022

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.

Original entry on oeis.org

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

Offset: 1

Michel Lagneau, Nov 30 2020

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).

			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.

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

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:

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

A350707 Numbers m such that all prime factors of m^2+1 are Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 18, 34, 57, 144, 239, 322, 610, 1134903170

Offset: 1

Michel Lagneau, Mar 27 2022

The Fibonacci numbers in the sequence include 1, 2, 3, 5, 8, 144, 610 and 1134903170.

The sequence includes terms of the form sqrt(f(n) - 1) and sqrt(5 * f(n) - 1), where f(n) = Fibonacci(A281087(n)) * Fibonacci(A281087(n)+2) = A140362(n). - Daniel Suteu, Mar 29 2022

			57 is in the sequence because 57^2+1 = 2*5^3*13 and 2, 5 and 13 are Fibonacci numbers;
1134903170 = Fibonacci(45) is in the sequence because 1134903170^2+1 = 433494437*2971215073 = Fibonacci(43)*Fibonacci(47).

The sequence contains A281618 and A285282.

Cf. A000032, A000045, A005478, A352290, A245236, A339461.

with(numtheory):
A005478:={2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497,1066340417491710595814572169, 19134702400093278081449423917}:
for n from 0 to 11000 do:
   y:=factorset(n^2+1):n0:=nops(y):
   if A005478 intersect y = y
       then
       print(n):
       else
     fi:
od:

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(m) = my(f=factor(m^2+1)); for (i=1, #f~, if (!isfib(f[i,1]), return(0))); return(1); \\ Michel Marcus, Mar 29 2022

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A352290 Numbers m such that the greatest prime factor of m^2 + 1 is a Fibonacci number.

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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.

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A350707 Numbers m such that all prime factors of m^2+1 are Fibonacci numbers.

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