

A154257


Number of triples <p,s,t> such that p+F_s+F_t=n, where p is an odd prime, s and t are greater than one and the Fibonacci number F_s or F_t is odd.


20



0, 0, 0, 0, 1, 2, 3, 4, 6, 6, 7, 6, 7, 8, 6, 10, 8, 10, 10, 10, 12, 10, 10, 10, 12, 14, 13, 12, 15, 8, 12, 12, 13, 14, 13, 10, 16, 10, 13, 16, 11, 16, 11, 14, 17, 16, 15, 12, 12, 16, 11, 20, 13, 14, 13, 12, 12, 18, 12, 16, 14, 14, 19, 16, 18, 20, 16, 18, 15, 18, 16, 12, 16, 18, 19, 22, 18
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OFFSET

1,6


COMMENTS

On Dec 23 2008, _ZhiWei_ Sun made a conjecture that states that a(n)>0 for all n=5,6,... (i.e., any integer n>4 can be written as the sum of an odd prime, an odd Fibonacci number and a positive Fibonacci number). This has been verified for n up to 10^14 by D. S. McNeil; the conjecture looks more difficult than the Goldbach conjecture since Fibonacci numbers are much more sparse than prime numbers. Sun also conjectured that c=lim inf_n a(n)/log n is greater than 2 and smaller than 3.
ZhiWei Sun has offered a monetary reward for settling this conjecture.


REFERENCES

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103107.
Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183190.


LINKS

ZhiWei Sun, Table of n, a(n), n=1..50000.
D. S. McNeil, Sun's strong conjecture
ZhiWei Sun, A promising conjecture: n=p+F_s+F_t
ZhiWei Sun, A summary concerning my conjecture n=p+F_s+F_t (II)
K. J. Wu and Z.W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m2^n and x^2F_{3n}/2, Math. Comp. 78 (2009) 1853, [DOI], arXiv:math.NT/0702382


EXAMPLE

For n=9 the a(9)=6 solutions are 3 + F_4 + F_4, 3 + F_2 + F_5, 3 + F_5 + F_2, 5 + F_3 + F_3, 5 + F_2 + F_4, 5 + F_4 + F_2.


MATHEMATICA

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[(Mod[n, 2]==0Mod[x, 3]>0)&&PQ[nFibonacci[x]Fibonacci[y]], 1, 0], {x, 2, 2*Log[2, Max[2, n]]}, {y, 2, 2*Log[2, Max[2, nFibonacci[x]]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]


CROSSREFS

Cf. A000040, A000045, A156695. See A144559 for another version.
Sequence in context: A214322 A075527 A325869 * A297351 A060019 A093451
Adjacent sequences: A154254 A154255 A154256 * A154258 A154259 A154260


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 05 2009


EXTENSIONS

The new verification record is 10^14 (due to D. S. McNeil).  ZhiWei Sun, Jan 17 2009


STATUS

approved



