A155904 Number of ways to write 2n-1 as p+2^x+5*2^y with p an odd prime and x,y positive integers.

0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 4, 3, 5, 6, 4, 5, 4, 4, 6, 5, 6, 7, 7, 5, 7, 11, 5, 10, 8, 5, 10, 7, 5, 8, 8, 7, 6, 10, 6, 8, 13, 9, 12, 10, 8, 14, 10, 7, 13, 12, 7, 10, 10, 9, 10, 17, 8, 11, 11, 9, 16, 12, 7, 13, 8, 10, 7, 8, 10, 11, 14, 5, 14, 14, 10, 17, 12, 7, 11, 12, 10, 12, 10, 12, 13, 17

Offset: 1

Author

Zhi-Wei Sun, Jan 30 2009

Keywords

• nice
• nonn

Comments

On Jan 21 2009, Zhi-Wei Sun conjectured that a(n)>0 for n=8,9,...; in other words, any odd integer m>=15 can be written as the sum of an odd prime, a positive power of 2 and five times a positive power of 2. Sun has verified this for odd integers m<10^8. As 5*2^y=2^y+2^{y+2}, the conjecture implies that each odd integer m>8 can be written as the sum of an odd prime and three positive powers of two. [It is known that there are infinitely many positive odd integers not of the form p+2^x+2^y (R. Crocker, 1971).] Sun also conjectured that there are infinitely many positive integers n with a(n)=a(n+1); here is the list of such positive integers n: 1, 2, 3, 4, 5, 6, 9, 10, 11, 19, 24, 36, 54, 60, 75, 90, 98, 101, 105, 135, 153, 173, ...

Examples

			For n=15 the a(15)=5 solutions are 29 = 17 + 2 + 5*2 = 11 + 2^3 + 5*2 = 3 + 2^4 + 5*2 = 7 + 2 + 5*2^2 = 5 + 2^2 + 5*2^2.
		

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) for n = 1..50000
• D. S. McNeil, Various and sundry (a report on Sun's conjectures)
• 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
• Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075
• 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.

Crossrefs

Cf. A000040, A000079, A155860, A154257, A154285, A155114, A154536, A154404, A154940.

Programs

Mathematica

PQ[x_]:=x>2&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-5*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/5]},{y,1,Log[2,2n-1-5*2^x]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]

Formula

a(n) = |{: p+2^x+5*2^y=2n-1 with p an odd prime and x,y positive integers}|.