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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
OFFSET
1,6
COMMENTS
On Dec 23 2008, _Zhi-Wei_ 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.
Zhi-Wei 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), 103-107.
Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.
LINKS
K. J. Wu and Z.-W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{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]==0||Mod[x, 3]>0)&&PQ[n-Fibonacci[x]-Fibonacci[y]], 1, 0], {x, 2, 2*Log[2, Max[2, n]]}, {y, 2, 2*Log[2, Max[2, n-Fibonacci[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 * A336406 A297351 A060019
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 05 2009
EXTENSIONS
The new verification record is 10^14 (due to D. S. McNeil). - Zhi-Wei Sun, Jan 17 2009
STATUS
approved