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

1,1

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

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

Example

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

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

Crossrefs

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

Sequence in context: A368790 A048801 A152915 * A071300 A062395 A352986

Adjacent sequences: A168380 A168381 A168382 * A168384 A168385 A168386

Keyword

base,easy,nonn,changed

Author

Jason Earls, Nov 24 2009

Extensions

Definition clarified by Robert Israel, Aug 22 2024

Status

approved