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A132145 Numbers that can be presented as a sum of a prime number and a Fibonacci number (0 is considered to be a Fibonacci number). 3

%I #16 Aug 04 2015 01:06:22

%S 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,

%T 27,28,29,30,31,32,33,34,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,

%U 51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72

%N Numbers that can be presented as a sum of a prime number and a Fibonacci number (0 is considered to be a Fibonacci number).

%C This sequence is the union of prime numbers and sequence A132147. It is also the complement of A132144.

%C Lee shows that the set of the numbers that are the sum of a prime and a Fibonacci number has positive lower asymptotic density. [_Jonathan Vos Post_, Nov 02 2010]

%H T. D. Noe, <a href="/A132145/b132145.txt">Table of n, a(n) for n=1..1000</a>

%H K. S. Enoch Lee, <a href="http://arxiv.org/abs/1011.0173">On the sum of a prime and a Fibonacci number</a>, arXiv:1011.0173 [math.NT], 2010.

%e 11 = 3+8, the sum of a prime number (3) and a Fibonacci number (8).

%p N:= 1000: # for all entries <= N

%p Primes:= select(isprime,{$1..N}):

%p phi:= (1+sqrt(5))/2:

%p Fibs:= {seq(combinat:-fibonacci(i),i=0..floor(log[phi]((N+1)*sqrt(5))))}:

%p sort(convert(select(`<=`,{seq(seq(f+p,f=Fibs),p=Primes)},N),list)); # _Robert Israel_, Aug 03 2015

%t Take[Union[Flatten[Table[Fibonacci[n] + Prime[k], {n, 70}, {k, 70}]], Table[Prime[k], {k, 70}]], 70]

%Y Cf. A132144, A132147.

%K nonn

%O 1,1

%A _Tanya Khovanova_, Aug 12 2007

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