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

1,9

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, ...

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.

Formula

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A341584 A110884 A303637 * A241514 A125913 A122386

Adjacent sequences: A155901 A155902 A155903 * A155905 A155906 A155907

Keyword

nice,nonn

Author

Zhi-Wei Sun, Jan 30 2009

Status

approved