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A115230
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Let p = prime(n); a(n) = number of ways to write p = 2^i + q^j where i >= 0, j >= 1, q = odd prime.
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5
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1, 1, 2, 2, 3, 3, 3, 3, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 4, 3, 2, 2, 2, 2, 2, 4, 1, 3, 3, 4, 0, 2, 3, 1, 3, 3, 1, 4, 1, 1, 2, 4, 2, 1, 3, 3, 2, 1, 3, 1, 3, 2, 1, 3, 2, 2, 3, 4, 2, 1, 2, 2, 0, 1, 3, 2, 4, 2, 2, 0, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 3, 0, 2, 3, 2, 1, 1, 3, 1, 4
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OFFSET
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1,3
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LINKS
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Table of n, a(n) for n=1..105.
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FORMULA
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a(n) = sum(A036987(k-1)*A000035(p-k)*A010055(p-k): 1<=k<p, p=prime(n)). [Reinhard Zumkeller, Apr 29 2010]
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EXAMPLE
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n=25: A000040(25) = 97 = 2^6+3*11 = 2^5+5*13 = 2^4+3^4 = 2^3+89^1 = 2^2+3*31 = 2^1+5*19 = 2^0+3*2^5, therefore a(25)=#{[16+81],[8+89]}=2.
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MAPLE
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From Reinhard Zumkeller, Apr 30 2010: (Start)
A000035 := proc(n) n mod 2 ; end proc:
A000108 := proc(n) binomial(2*n, n)/(n+1) ; end proc:
A036987 := proc(n) A000108(n) mod 2 ; end proc:
A010055 := proc(n) if n = 1 then 1; else numtheory[factorset](n) ; if nops(%) = 1 then 1; else 0; end if; end if: end proc:
A115230 := proc(n) p := ithprime(n) ; add(A036987(k-1)*A000035(p-k)*A010055(p-k), k=1..p-1) ; end proc: seq(A115230(n), n=1..40) ; # R. J. Mathar, Apr 30 2010 (End)
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MATHEMATICA
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f[p_] := Length@ Table[q = p - 2^exp; If[ PrimeNu@ q == 1, {q}, Sequence @@ {}], {exp, 0, Floor@ Log2@ p}]; Table[ f[ Prime[ n]], {n, 105}] (* Robert G. Wilson v, Oct 05 2014 *)
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CROSSREFS
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Cf. A115231-A115233, A000079, A061345.
Sequence in context: A321859 A321860 A266123 * A304733 A165024 A300402
Adjacent sequences: A115227 A115228 A115229 * A115231 A115232 A115233
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller, Jan 17 2006
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EXTENSIONS
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Recomputed by Charles R Greathouse IV, Ray Chandler, R. J. Mathar, and Reinhard Zumkeller, Apr 29 2010; thanks to Charles R Greathouse IV, who pointed out that there were many errors in entries of A115230-A115233.
Edited by N. J. A. Sloane, Apr 30 2010
Formula corrected, thanks to R. J. Mathar who found an error in it Reinhard Zumkeller, Apr 30 2010
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STATUS
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approved
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