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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.

%I A168383 #13 Aug 23 2024 10:50:14
%S A168383 2,9,65,77,93,95,123,323,335,343,377,395,415,425,437,527,545,553,583,
%T A168383 586,670,700,715,723,726,731,749,783,801,804,833,838,849,851,901,903,
%U A168383 905,906,923,957,959,964,965,1003,1078,1081,1113,1115
%N 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.
%C A168383 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
%D A168383 J. Earls, "Fibonacci Prime Decompositions," Mathematical Bliss, Pleroma Publications, 2009, pages 76-79. ASIN: B002ACVZ6O
%H A168383 Robert Israel, <a href="/A168383/b168383.txt">Table of n, a(n) for n = 1..10000</a>
%e A168383 In the decomposition of 1081, the prime and Fibonacci both have three digits: 1081 = 144 + 937.
%p A168383 filter:= proc(n) local f,i,d,state;
%p A168383   state:= 0;
%p A168383   for i from 0 do
%p A168383     f:= combinat:-fibonacci(i);
%p A168383     if f >= n then return (state = 1) fi;
%p A168383     if isprime(n-f) then
%p A168383       state:= state+1;
%p A168383       if state = 2 then return false fi;
%p A168383       if f = 0 then d:= 1 else d:= 1+ilog10(f) fi;
%p A168383       if 1+ilog10(n-f) <> d then return false fi;
%p A168383     fi
%p A168383   od;
%p A168383 end proc:
%p A168383 select(filter, [$1..2000]); # _Robert Israel_, Aug 22 2024
%Y A168383 Cf. A000045, A132144, A375642.  Contained in A375643.
%K A168383 base,easy,nonn
%O A168383 1,1
%A A168383 _Jason Earls_, Nov 24 2009
%E A168383 Definition clarified by _Robert Israel_, Aug 22 2024