

A154285


Number of ordered triples <p,s,t> satisfying p+L_s+L_t=n, where p is an odd prime, s and t are nonnegative and the Lucas number L_s or L_t is odd.


15



0, 0, 0, 0, 1, 2, 3, 6, 4, 8, 5, 8, 6, 8, 9, 12, 10, 12, 9, 10, 12, 14, 9, 14, 12, 14, 10, 14, 8, 10, 10, 16, 11, 16, 12, 18, 12, 16, 10, 12, 13, 16, 15, 16, 13, 14, 13, 16, 14, 18
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OFFSET

1,6


COMMENTS

ZhiWei Sun conjectured that a(n)>0 for all n=5,6,...; in other words, any integer n>4 can be represented as the sum of an odd prime, an odd Lucas number and a Lucas number. This has been verified up to 1.5*10^8. Sun thought that the constant lim inf_n a(n)/log(n) is greater than 2 and smaller than 3. For k=2,3 Sun also conjectured that any integer n>4 can be written in the form p+L_s+(L_t)^k, where p is an odd prime and L_s or L_t is odd.
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 summary concerning my 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, arXiv:math.NT/0702382


EXAMPLE

For n=3 the a(7)=4 solutions are 3+L_1+L_2, 3+L_2+L_1, 5+L_1+L_1.


MATHEMATICA

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


CROSSREFS

Cf. A000032, A000040, A154257, A154258, A154263, A156695.
Sequence in context: A138728 A291604 A082332 * A258078 A036552 A257987
Adjacent sequences: A154282 A154283 A154284 * A154286 A154287 A154288


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Jan 06 2009


STATUS

approved



