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A154258 Number of triples <p,s,t> such that p+F_s+(F_t)^2=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. 4
0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 2, 4, 3, 5, 4, 5, 6, 3, 5, 6, 5, 7, 4, 5, 7, 4, 6, 7, 6, 6, 6, 5, 11, 6, 8, 6, 6, 7, 6, 9, 9, 4, 9, 5, 9, 10, 6, 8, 8, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

Zhi-Wei Sun conjectured 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, a positive Fibonacci number and a square of a positive Fibonacci number, with one of the two Fibonacci numbers odd). He has verified this for n up to 3*10^7.

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 summary concerning my 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., in press. arXiv:math.NT/0702382

EXAMPLE

For n=10 the a(10)=3 solutions are 3+F_4+(F_3)^2, 5+F_2+(F_3)^2, 7+F_3+(F_2)^2.

MATHEMATICA

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

CROSSREFS

Cf. A000040, A000045, A154257

Sequence in context: A276273 A039643 A288887 * A253900 A327487 A105496

Adjacent sequences:  A154255 A154256 A154257 * A154259 A154260 A154261

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jan 05 2009

STATUS

approved

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Last modified April 8 12:41 EDT 2020. Contains 333314 sequences. (Running on oeis4.)