A157218 Number of ways to write the n-th positive odd integer in the form p+2^x+7*2^y with p a prime congruent to 1 mod 6 and x,y positive integers.

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

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 for all n=18,19,...; in other words, any odd integer greater than 34 can be written as the sum of a prime congruent to 1 mod 6, a positive power of 2 and seven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7, and Qing-Hu Hou continued the verification for odd integers below 1.5*10^8 (on Sun's request). Compare the conjecture with R. 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=19 the a(19)=3 solutions are 2*19 - 1 = 7 + 2 + 7*2^2 = 7 + 2^4 + 7*2 = 19 + 2^2 + 7*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..200000
• Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075
• 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 and M.-H. Le, Integers not of the form c*(2^a + 2^b) + p^{alpha}, Acta Arith. 99(2001), 183-190.

Crossrefs

A000040, A000079, A155860, A155904, A156695, A154257, A154285, A155114, A154536, A154404, A154940.

Programs

Mathematica

PQ[x_]:=x>1&&Mod[x,6]==1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-7*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/7]},{y,1,Log[2,Max[2,2n-1-7*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

Formula

a(n) = |{: p+2^x+7*2^y=2n-1 with p a prime congruent to 1 mod 6 and x,y positive integers}|.