A168383 Numbers expressible as the sum of a prime and a Fibonacci number in only one way, and such that the prime and Fibonacci number have the same number of decimal digits.

2, 9, 65, 77, 93, 95, 123, 323, 335, 343, 377, 395, 415, 425, 437, 527, 545, 553, 583, 586, 670, 700, 715, 723, 726, 731, 749, 783, 801, 804, 833, 838, 849, 851, 901, 903, 905, 906, 923, 957, 959, 964, 965, 1003, 1078, 1081, 1113, 1115

Offset: 1

Author

Jason Earls, Nov 24 2009

Keywords

• base
• easy
• nonn

Comments

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. - Robert Israel, Aug 22 2024

Examples

			In the decomposition of 1081, the prime and Fibonacci both have three digits: 1081 = 144 + 937.
		

References

• J. Earls, "Fibonacci Prime Decompositions," Mathematical Bliss, Pleroma Publications, 2009, pages 76-79. ASIN: B002ACVZ6O

Links

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

Crossrefs

Cf. A000045, A132144, A375642. Contained in A375643.

Programs

Maple

filter:= proc(n) local f,i,d,state;
  state:= 0;
  for i from 0 do
    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;
      if f = 0 then d:= 1 else d:= 1+ilog10(f) fi;
      if 1+ilog10(n-f) <> d then return false fi;
    fi
  od;
end proc:
select(filter, [$1..2000]); # Robert Israel, Aug 22 2024

Extensions

Definition clarified by Robert Israel, Aug 22 2024