A154290 Number of ordered triples <p,s,t> satisfying p+F_s+L_t = n, where p is an odd prime, s >= 2 and F_s or L_t is odd.

0, 0, 0, 0, 1, 2, 3, 5, 5, 7, 6, 8, 6, 8, 8, 10, 9, 9, 11, 11, 10, 14, 10, 11, 11, 15, 13, 14, 10, 10, 11, 12, 12, 14, 15, 14, 13, 14, 12, 13, 11, 16, 13, 15, 15, 16, 13, 17, 12, 17

Offset

1,6

Comments

Zhi-Wei Sun conjectured that a(n)>0 for every n=5,6,...; in other words, any integer n>4 can be written as the sum of an odd prime, a positive Fibonacci number and a Lucas number, with the Fibonacci number or the Lucas number odd. Moreover, Sun conjectured that lim inf_n a(n)/log(n) is greater than 3 and smaller than 4.

References

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

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

Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.

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, arXiv:math.NT/0702382, Math. Comp. 78 (2009) 1853-1868.

Example

For n=10 the a(10)=7 solutions are 3+F_4+L_3, 3+F_5+L_0, 5+F_2+L_3, 5+F_3+L_2, 5+F_4+L_0, 7+F_2+L_0, 7+F_3+L_1.

Mathematica

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

Crossrefs

Cf. A000040, A000045, A000032, A154257, A154285

Sequence in context: A081836 A263721 A351259 * A267259 A002334 A115732

Adjacent sequences: A154287 A154288 A154289 * A154291 A154292 A154293

Keyword

nonn

Author

Zhi-Wei Sun, Jan 06 2009, Jan 07 2008

Status

approved