A155860 Number of ways to write 2n-1 as p + 2^x + 3*2^y with p an odd prime and x,y positive integers.

0, 0, 0, 0, 0, 1, 2, 2, 3, 4, 5, 3, 5, 7, 4, 7, 9, 5, 6, 9, 5, 7, 11, 6, 6, 12, 5, 9, 13, 8, 10, 12, 4, 11, 15, 6, 10, 15, 5, 9, 16, 9, 9, 17, 8, 8, 17, 8, 10, 16, 8, 11, 13, 10, 10, 20, 7, 12, 23, 10, 10, 21, 9, 11, 18, 11, 8, 18, 9, 11, 20, 9, 13, 17, 9, 12, 19, 9, 13, 22, 6, 13, 21, 10, 10, 21

Offset: 1

Author

Zhi-Wei Sun, Jan 29 2009

Keywords

• nice
• nonn

Comments

On Jan 21 2009, Zhi-Wei Sun conjectured that a(n)>0 for n=6,7,...; in other words, any odd integer m>10 can be written as the sum of an odd prime, a positive power of 2 and three times a positive power of 2. Sun verified this for odd integers m<10^7. On Sun's request, Qing-Hu Hou and Charles R Greathouse IV continued the verification for odd integers below 2*10^8 and 10^10 respectively and found no counterexamples.

As 3*2^y = 2^y + 2^{y+1}, Sun's conjecture implies that each odd integer m>8 can be written as the sum of an odd prime and three positive powers of two. Note that Paul Erdős asked whether there is a positive integer r such that every odd integer m>3 can be written as the sum of a prime and at most r powers of 2.

Zhi-Wei Sun also raised the following problem: For k=3,5,...,61 determine whether any odd integer m>2k+3 can be written in the form p + 2^x + k*2^y with p an odd prime and x,y positive integers. Sun observed that 353 is not of the form p + 2^x + 51*2^y and Qing-Hu Hou continued the search for m<2.5*10^7 and found that 22537515 is not of the form p + 2^x + 47*2^y. For k=3,5,...,45,49,53,55,...,61, Sun has checked odd integers below 10^8 and found no odd integer m>2k-3 not of the form p + 2^x + k*2^y.

Examples

			For n=10 the a(10)=4 solutions are 19 = 3 + 2^2 + 3*2^2 = 5 + 2 + 3*2^2 = 5 + 2^3 + 3*2 = 11 + 2 + 3*2.
		

References

• R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
• Z.-W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..50000
• Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61
• Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
• D. S. McNeil, Various and sundry (a report on Sun's conjectures)
• Z.-W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075

Crossrefs

Cf. A154257, A154285, A155114, A154536, A154404, A154940.

Programs

Mathematica

PQ[x_]:=x>2&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-3*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/3]},{y,1,Log[2,Max[2,2n-1-3*2^x]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]

Formula

a(n) = |{: p+2^x+3*2^y = 2n-1 with p an odd prime and x,y positive integers}|.