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

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

Offset

1,3

Comments

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

Links

Table of n, a(n) for n=1..15.

Example

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).

Maple

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:

Prog

(PARI) 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

Crossrefs

The sequence contains A281618 and A285282.

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

Sequence in context: A039892 A278645 A352290 * A285282 A105404 A238378

Adjacent sequences: A350704 A350705 A350706 * A350708 A350709 A350710

Keyword

nonn,hard,more

Author

Michel Lagneau, Mar 27 2022

Status

approved