A168383 -id:A168383 - cp's OEIS frontend

A375642 a(n) is the number of i for which n - Fibonacci(i) is prime.

0, 1, 3, 3, 3, 3, 3, 4, 1, 3, 2, 3, 3, 3, 3, 3, 1, 4, 3, 4, 2, 2, 2, 5, 2, 3, 1, 2, 1, 3, 3, 5, 1, 3, 0, 3, 3, 3, 3, 2, 2, 4, 2, 5, 3, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 1, 4, 3, 5, 2, 3, 1, 3, 2, 4, 2, 1, 2, 5, 2, 6, 3, 2, 1, 2, 2, 4, 3, 2, 1, 5, 1, 3, 2, 2, 1, 2, 3, 5, 1, 3, 1, 3, 2, 3, 1

Offset: 1

Robert Israel, Aug 22 2024

nonn

			a(5) = 3 because 5 - Fibonacci(0) = 5, 5 - Fibonacci(3) = 3 and 5 - Fibonacci(4) = 2 are prime.

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

Cf. A000045, A108852, A132144, A168383, A169791.

fcount:= proc(n) local f,i,d,c;
  c:= 0;
  for i from 0 do
    f:= combinat:-fibonacci(i);
    if f >= n then return c fi;
    if isprime(n-f) then
      c:= c+1;
    fi
  od;
end proc:
map(f, [$1..200]);

a[n_]:=Sum[Boole[PrimeQ[n-Fibonacci[i]]],{i,Select[Range[0,n],n>Fibonacci[#]&]}]; Array[a,99] (* Stefano Spezia, Aug 23 2024 *)

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

Original entry on oeis.org

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

Robert Israel, Aug 22 2024

nonn

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.

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

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

Cf. A000045, A132144, A375642. Contains A168383.

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]);

cp's OEIS Frontend

A375642 a(n) is the number of i for which n - Fibonacci(i) is prime.

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