A157372 Number of ways to write the (n+50)-th positive odd integer in the form p+2^x+51*2^y with p an odd prime and x,y positive integers.

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

Offset: 1

Author

Zhi-Wei Sun, Feb 28 2009

Keywords

• nice
• nonn

Comments

Zhi-Wei Sun guessed that a(n)=0 if and only if n=1,2,3,127; in other words, except for 353, any odd integer greater than 105 can be written as the sum of an odd prime, a positive power of two and 51 times a positive power of two. D. S. McNeil has verified this for odd integers below 10^12. This is a part of the project for the form p+2^x+k*2^y with k=3,5,...,61 initiated by Zhi-Wei Sun in Jan. 2009.

Examples

			For n = 9 the a(9) = 3 solutions are: 2*59-1 = 7+2^3+51*2 = 11+2^2+51*2 = 13+2+51*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..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 curious conjecture about p+2^x+11*2^y
• Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075

Crossrefs

Cf. A000040, A000079, A157237, A157242, A155860, A155904, A156695, A154257, A154285, A155114, A154536.

Programs

Mathematica

PQ[x_]:=x>1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2(n+50)-1-51*2^x-2^y],1,0], {x,1,Log[2,(2(n+50)-1)/51]},{y,1,Log[2,Max[2,2(n+50)-1-51*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

Formula

a(n)=|{: p+2^x+51*2^y=2(n+50)-1 with p an odd prime and x,y positive integers}|