|
%I #18 Oct 05 2014 21:56:21
%S 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,
%T 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,
%U 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
%N Let p = prime(n); a(n) = number of ways to write p = 2^i + q^j where i >= 0, j >= 1, q = odd prime.
%F a(n) = sum(A036987(k-1)*A000035(p-k)*A010055(p-k): 1<=k<p, p=prime(n)). [_Reinhard Zumkeller_, Apr 29 2010]
%e 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.
%p From _Reinhard Zumkeller_, Apr 30 2010: (Start)
%p A000035 := proc(n) n mod 2 ; end proc:
%p A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
%p A036987 := proc(n) A000108(n) mod 2 ; end proc:
%p 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:
%p 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)
%t 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 *)
%Y Cf. A115231-A115233, A000079, A061345.
%K nonn
%O 1,3
%A _Reinhard Zumkeller_, Jan 17 2006
%E 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.
%E Edited by _N. J. A. Sloane_, Apr 30 2010
%E Formula corrected, thanks to _R. J. Mathar_ who found an error in it _Reinhard Zumkeller_, Apr 30 2010
|
