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A155860
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Number of ways to write 2n-1 as p + 2^x + 3*2^y with p an odd prime and x,y positive integers.
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7
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0, 0, 0, 0, 0, 1, 2, 2, 3, 4, 5, 3, 5, 7, 4, 7, 9, 5, 6, 9, 5, 7, 11, 6, 6, 12, 5, 9, 13, 8, 10, 12, 4, 11, 15, 6, 10, 15, 5, 9, 16, 9, 9, 17, 8, 8, 17, 8, 10, 16, 8, 11, 13, 10, 10, 20, 7, 12, 23, 10, 10, 21, 9, 11, 18, 11, 8, 18, 9, 11, 20, 9, 13, 17, 9, 12, 19, 9, 13, 22, 6, 13, 21, 10, 10, 21
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OFFSET
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1,7
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COMMENTS
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On Jan 21 2009, Zhi-Wei Sun conjectured that a(n)>0 for n=6,7,...; in other words, any odd integer m>10 can be written as the sum of an odd prime, a positive power of 2 and three times a positive power of 2. Sun verified this for odd integers m<10^7. On Sun's request, Qing-Hu Hou and Charles R Greathouse IV continued the verification for odd integers below 2*10^8 and 10^10 respectively and found no counterexamples.
As 3*2^y = 2^y + 2^{y+1}, Sun's conjecture implies that each odd integer m>8 can be written as the sum of an odd prime and three positive powers of two. Note that Paul Erdős asked whether there is a positive integer r such that every odd integer m>3 can be written as the sum of a prime and at most r powers of 2.
Zhi-Wei Sun also raised the following problem: For k=3,5,...,61 determine whether any odd integer m>2k+3 can be written in the form p + 2^x + k*2^y with p an odd prime and x,y positive integers. Sun observed that 353 is not of the form p + 2^x + 51*2^y and Qing-Hu Hou continued the search for m<2.5*10^7 and found that 22537515 is not of the form p + 2^x + 47*2^y. For k=3,5,...,45,49,53,55,...,61, Sun has checked odd integers below 10^8 and found no odd integer m>2k-3 not of the form p + 2^x + k*2^y.
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REFERENCES
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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.
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LINKS
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FORMULA
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a(n) = |{<p,x,y>: p+2^x+3*2^y = 2n-1 with p an odd prime and x,y positive integers}|.
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EXAMPLE
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For n=10 the a(10)=4 solutions are 19 = 3 + 2^2 + 3*2^2 = 5 + 2 + 3*2^2 = 5 + 2^3 + 3*2 = 11 + 2 + 3*2.
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MATHEMATICA
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PQ[x_]:=x>2&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-3*2^x-2^y], 1, 0], {x, 1, Log[2, (2n-1)/3]}, {y, 1, Log[2, Max[2, 2n-1-3*2^x]]}] Do[Print[n, " ", RN[n]]; Continue, {n, 1, 50000}]
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CROSSREFS
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KEYWORD
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nice,nonn
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AUTHOR
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STATUS
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approved
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