The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A154421 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number. 2
0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 2, 5, 2, 4, 3, 4, 4, 5, 2, 6, 2, 7, 5, 7, 3, 9, 3, 9, 4, 7, 3, 6, 4, 9, 3, 10, 3, 8, 4, 6, 5, 8, 6, 8, 3, 9, 4, 8, 6, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
On Jan 09 2009, Zhi-Wei Sun conjectured that a(n)>0 for all n=6,7,.... ; in other words, any integer n>5 can be written in the form p+F_s+L_{3t} with p an odd prime, s positive and t nonnegative. [Compare this with the conjecture related to the sequence A154290.] Sun verified the above conjecture up to 5*10^6 and Qing-Hu Hou continued the verification up to 2*10^8. If we set v_0=2, v_1=4 and v_{n+1}=4v_n+v_{n-1} for n=1,2,3,..., then L_{3t}=v_t is at least 4^t for every t=0,1,2,.... On Jan 17 2009, D. S. McNeil found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.
REFERENCES
R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
LINKS
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, Math. Comp. 78 (2009) 1853, [DOI], arXiv:math.NT/0702382
FORMULA
a(n) = |{<p,s,t>: p+F_s+L_{3t}=n with p an odd prime, s>1 and t nonnegative}|.
EXAMPLE
For n=8 the a(8)=3 solutions are 3 + F_4 + L_0, 3 + F_2 + L_3, 5 + F_2 + L_0.
MATHEMATICA
PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n-2*Fibonacci[3x+1]+Fibonacci[3x]-Fibonacci[y]], 1, 0], {x, 0, Log[2, n]}, {y, 2, 2*Log[2, Max[2, n-2*Fibonacci[3x+1]+Fibonacci[3x]]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]
CROSSREFS
Sequence in context: A127750 A112528 A366687 * A057034 A075410 A023513
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 09 2009
EXTENSIONS
McNeil's counterexample added by Zhi-Wei Sun, Jan 20 2009
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 13 22:21 EDT 2024. Contains 373391 sequences. (Running on oeis4.)