A154536 Positive integers that can be written as the sum of a positive Pell number and twice a positive Pell number.

3, 4, 5, 6, 7, 9, 11, 12, 14, 15, 16, 22, 25, 26, 29, 31, 33, 36, 39, 53, 59, 60, 63, 70, 72, 74, 80, 87, 94, 128, 141, 142, 145, 152, 169, 171, 173, 179, 193, 210, 227, 309, 339, 340, 343, 350, 367, 408, 410, 412

Offset

1,1

Comments

On Jan 10 2009, Zhi-Wei Sun conjectured that any integer greater than 5 can be expressed as the sum of an odd prime and a term in the above sequence; in other words, each n=6,7,... can be written in the form p+P_s+2*P_t with p an odd prime and s,t>0. This has been verified up to 5*10^13 by D. S. McNeil (from London Univ.). Motivated by this conjecture, Qing-Hu Hou (from Nankai Univ.) observed and Zhi-Wei Sun proved that each term a(n) in the above sequence can be uniquely written in the form P_s+2P_t with s,t>0. Sun noted that 2176 cannot be written as the sum of a prime and two Pell numbers; D. S. McNeil found that 393185153350 cannot be written in the form p+P_s+3P_t and 872377759846 cannot be written in the form p+P_s+4P_t, where p is a prime and s and t are nonnegative.

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.

Links

Zhi-Wei SUN, Table of n, a(n), n=1..179.

D. S. McNeil, Sun's strong conjecture

D. S. McNeil, Various and sundry: a report on Sun's conjectures

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, Math. Comp., in press. arXiv:math.NT/0702382

Example

For n=12 the a(12)=22 solution is 22 = P_4 + 2*P_3.

Mathematica

P[n_]:=P[n]=2*P[n-1]+P[n-2] P[0]=0 P[1]=1 i:=0 Do[Do[If[n==2*P[x]+P[y], i=i+1; Print[i, " ", n]], {x, 1, Max[1, Log[2, n]]}, {y, 1, Log[2, n]+1}]; Continue, {n, 1, 100000}]

Crossrefs

Cf. A000129, A000040, A154257, A154285, A154364, A154417, A154421, A156695.

Sequence in context: A048869 A039051 A047564 * A298110 A091815 A081692

Adjacent sequences: A154533 A154534 A154535 * A154537 A154538 A154539

Keyword

nonn

Author

Zhi-Wei Sun, Jan 11 2009

Extensions

Mentioned McNeil's verification record for the representation n = p + P_s + 2P_t and his examples for n not of the form p + P_s + 3P_t and n not of the form p + P_s + 4P_t. - Zhi-Wei Sun, Jan 17 2009

D. S. McNeil has verified the conjecture up to 5*10^13. - Zhi-Wei Sun, Jan 20 2009

Status

approved