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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A029250 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

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Last modified December 15 11:43 EST 2019. Contains 329999 sequences. (Running on oeis4.)