A375643 Numbers that are the sum of a prime and a Fibonacci number in exactly one way.

2, 9, 17, 27, 29, 33, 59, 65, 70, 77, 83, 85, 89, 90, 93, 95, 99, 121, 123, 124, 127, 129, 133, 143, 145, 146, 153, 155, 166, 169, 174, 179, 188, 189, 190, 195, 203, 210, 217, 219, 222, 237, 239, 243, 249, 258, 261, 267, 269, 289, 297, 302, 303, 305, 308, 309, 310, 321, 323, 327, 331, 333, 335

Offset: 1

Author

Robert Israel, Aug 22 2024

Keywords

• nonn

Comments

Numbers k such that k - A000045(i) is prime for exactly one i >= 0.

1 = Fibonacci(1) = Fibonacci(2), so cases where the Fibonacci number is 1 are counted as two ways. Also, if Fibonacci(i) and Fibonacci(j) are both primes (with i <> j), Fibonacci(i) + Fibonacci(j) and Fibonacci(j) + Fibonacci(i) are counted as two ways.

Examples

			a(5) = 29 is a term because 29 - Fibonacci(i) is prime only for i = 0.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000045, A132144, A375642. Contains A168383.

Programs

Maple

filter:= proc(n) local f,i,d,state;
  state:= 0;
  for i from 0 do
    if i = 2 then next fi;
    f:= combinat:-fibonacci(i);
    if f >= n then return (state = 1) fi;
    if isprime(n-f) then
      state:= state+1;
      if state = 2 then return false fi;
    fi
  od;
end proc:
select(filter, [$1..1000]);