A157242 Number of ways to write the n-th positive odd integer in the form p+2^x+11*2^y with p a prime congruent to 5 mod 6 and x,y positive integers.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 2, 2, 0, 2, 2, 1, 3, 2, 2, 3, 1, 2, 4, 2, 2, 5, 1, 2, 5, 2, 2, 4, 2, 2, 3, 2, 3, 4, 4, 3, 6, 2, 3, 6, 5, 1, 7, 4, 2, 6

Offset: 1

Author

Zhi-Wei Sun, Feb 25 2009

Keywords

• nice
• nonn

Comments

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if n<15 or n=17, 20, 23, 86, 124; in other words, except for 33, 39, 45, 171 and 247, any odd integer greater than 28 can be written as the sum of a prime p=5 (mod 6), a positive power of 2 and eleven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7. Knowing the conjecture from Sun, Qing-Hu Hou and D. S. McNeil have continued the verification for odd integers below 1.5*10^8 and 10^12 respectively, and they have found no counterexample. Compare the conjecture with Crocker's result that there are infinitely many positive odd integers not of the form p+2^x+2^y with p an odd prime and x,y positive integers.

Examples

			For n=18 the a(18)=2 solutions are 2*18-1=5+2^3+2*11=11+2+2*11.
		

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..200000
• Zhi-Wei Sun, A webpage: Mixed Sums of Primes and Other Terms, 2009.
• 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

Crossrefs

A000040, A000079, A157237, A155860, A155904, A156695, A154257, A154285, A155114, A154536

Programs

Mathematica

PQ[x_]:=x>1&&Mod[x,6]==5&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-11*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/11]},{y,1,Log[2,Max[2,2n-1-11*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

Formula

a(n)=|{: p+2^x+11*2^y=2n-1 with p a prime congruent to 5 mod 6 and x,y positive integers}|