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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 (list; graph; refs; listen; history; text; internal format)
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

Zhi-Wei Sun, Table of n, a(n), n=1..50000.

D. S. McNeil, Sun's strong conjecture

Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t

Zhi-Wei 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^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 * A297351 A060019 A093451

Adjacent sequences:  A154254 A154255 A154256 * A154258 A154259 A154260

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

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Last modified March 31 19:16 EDT 2020. Contains 333151 sequences. (Running on oeis4.)