login
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.
1
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
1,5
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.
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.
FORMULA
a(n)=|{<p,x,y>: p+2^x+51*2^y=2(n+50)-1 with p an odd prime and x,y positive integers}|
EXAMPLE
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.
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}]
KEYWORD
nice,nonn
AUTHOR
Zhi-Wei Sun, Feb 28 2009
STATUS
approved