A115230 Let p = prime(n); a(n) = number of ways to write p = 2^i + q^j where i >= 0, j >= 1, q = odd prime.

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

Offset: 1

Author

Reinhard Zumkeller, Jan 17 2006

Keywords

• nonn

Examples

			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.
		

Crossrefs

Cf. A115231-A115233, A000079, A061345.

Programs

Maple

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)

Mathematica

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 *)

Formula

a(n) = Sum_{k=1..prime(n)-1} A036987(k-1)*A000035(p-k)*A010055(p-k). - Reinhard Zumkeller, Apr 29 2010

Extensions

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) by Reinhard Zumkeller, Apr 30 2010