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

1,15

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.

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.

Formula

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

Example

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.

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}]

Crossrefs

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

Sequence in context: A167655 A346148 A262781 * A004718 A157225 A055347

Adjacent sequences: A157215 A157216 A157217 * A157219 A157220 A157221

Keyword

nice,nonn

Author

Zhi-Wei Sun, Feb 25 2009

Status

approved