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
KEYWORD
nice,nonn
AUTHOR
Zhi-Wei Sun, Feb 25 2009
STATUS
approved