

A155114


Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and twice a positive Fibonacci number.


8



0, 0, 0, 0, 0, 1, 1, 3, 2, 6, 3, 7, 3, 8, 5, 8, 6, 10, 5, 11, 6, 13, 7, 13, 7, 14, 5, 14, 7, 15, 8, 14, 4, 18, 8, 17, 7, 15, 5, 15, 11, 16, 8, 15, 7, 17, 12, 19, 10, 20, 10, 17, 10, 17, 13, 15, 11, 18, 8, 20, 10, 17, 9, 18, 11, 21, 11, 21, 7, 20, 11, 18, 11, 22, 9, 25, 11, 24, 13, 19, 14, 20, 11
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OFFSET

1,8


COMMENTS

Motivated by his conjecture related to A154257, on Dec 26 2008, ZhiWei Sun conjectured that a(n)>0 for n=6,7,... On Jan 15 2009, D. S. McNeil verified this up to 10^12 and found no counterexamples. See the sequence A154536 for another conjecture of this sort. Sun also conjectured that any integer n>7 can be written as the sum of an odd prime, twice a positive Fibonacci number and the square of a positive Fibonacci number; this has been verified up to 2*10^8.


REFERENCES

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103107.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..100000
D. S. McNeil, Various and sundry (a report on Sun's conjectures)
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)
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405413.
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) 18531866. arXiv:math.NT/0702382.


FORMULA

a(n) = {<p,s,t>: p+F_s+2F_t=n with p an odd prime and s,t>1}.


EXAMPLE

For n=10 the a(10)=6 solutions are 3 + F_4 + 2F_3, 3 + F_5 + 2F_2, 3 + F_2 + 2F_4, 5 + F_2 + 2F_3, 5 + F_4 + 2F_2, 7 + F_2 + 2F_2.


MATHEMATICA

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n2*Fibonacci[x]Fibonacci[y]], 1, 0], {x, 2, 2*Log[2, Max[2, n/2]]}, {y, 2, 2*Log[2, Max[2, n2*Fibonacci[x]]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 100000}]


CROSSREFS

Cf. A000040, A000045, A154257, A154258, A154263, A154285, A154290, A154417, A154536, A154404, A154940, A156695.
Sequence in context: A322907 A071018 A144559 * A038572 A245676 A060992
Adjacent sequences: A155111 A155112 A155113 * A155115 A155116 A155117


KEYWORD

nice,nonn


AUTHOR

ZhiWei Sun, Jan 20 2009


STATUS

approved



