login
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.
3
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
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
KEYWORD
base,easy,nonn
AUTHOR
Jason Earls, Nov 24 2009
EXTENSIONS
Definition clarified by Robert Israel, Aug 22 2024
STATUS
approved