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

1,5

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.

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

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}]

Crossrefs

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

Sequence in context: A358333 A237168 A350959 * A270559 A231727 A368876

Adjacent sequences: A157369 A157370 A157371 * A157373 A157374 A157375

Keyword

nice,nonn

Author

Zhi-Wei Sun, Feb 28 2009

Status

approved