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

1,3

Links

Table of n, a(n) for n=1..105.

Formula

a(n) = sum(A036987(k-1)*A000035(p-k)*A010055(p-k): 1<=k<p, p=prime(n)). [Reinhard Zumkeller, Apr 29 2010]

Example

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.

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 *)

Crossrefs

Cf. A115231-A115233, A000079, A061345.

Sequence in context: A321860 A348459 A266123 * A363937 A338686 A338687

Adjacent sequences: A115227 A115228 A115229 * A115231 A115232 A115233

Keyword

nonn

Author

Reinhard Zumkeller, Jan 17 2006

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 Reinhard Zumkeller, Apr 30 2010

Status

approved