A198255 - OEIS

%I #12 Sep 23 2018 20:58:55

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,

%T 0,0,1,0,0,0,2,0,2,0,0,0,2,0,2,0,0,0,2,0,1,0,1,0,2,0,5,0,0,0,1,0,3,1,

%U 2,0,3,0,4,1,0,1,3,0,5,0,2,0,4,0,1,2,2,0,3,0,4,2,2,1,3,0,6,1,2,0,6

%N Number of ways to write n as the sum of two coprime squarefree semiprimes.

%C This sequence is term by term less than or equal to A197629.  The first odd term where the inequality is strict is the 97th term.  The first even term that is strictly less than A197629 is the 248th term.

%C There are interesting bands in the scatterplot of this sequence. - _Antti Karttunen_, Sep 23 2018

%H Antti Karttunen, <a href="/A198255/b198255.txt">Table of n, a(n) for n = 1..65537</a>

%e Same as A197629 until a(97) since 97=(2)(3)(7)+(5)(11) and thus the value of the 97th term of A197629 is one greater.  The 248th term of A197629 is the first even term which is one greater since 248=(3)(5)(7)+(11)(13).

%e From _Antti Karttunen_, Sep 23 2018: (Start)

%e For n = 97 there are following six solutions: 97 = 6+91 = 10+87 = 15+82 = 35+62 = 39+58 = 46+51, thus a(97) = 6.

%e For n = 248 there are following seven solutions: 248 = 33+215 = 35+213 = 39+209 = 65+183 = 87+161 = 115+133 = 119+129, thus a(248) = 7.

%e (End)

%o (MATLAB)

%o function [asubn] = sps(n)

%o % Returns the number of coprime, squarefree, semiprime, partitions a+b of n.

%o r = 0; % r is the number of sps's of n

%o k=6;

%o while k < n/2,

%o     if gcd(k,n-k)==1

%o          if length(factor(k)) == 2

%o              if length(factor(n-k)) == 2

%o                  if prod(diff(factor(k)))*prod(diff(factor(n-k))) > 0

%o                     r = r + 1;

%o                  end

%o              end

%o          end

%o     end

%o     k = k + 1;

%o end

%o asubn = r;

%o end

%o (PARI) A198255(n) = sum(k=4, (n-1)\2, gcd(k, n-k)==1&&(2==bigomega(k))&&(2==bigomega(n-k))&&issquarefree(k)&&issquarefree(n-k)); \\ _Antti Karttunen_, Sep 23 2018, after _Charles R Greathouse IV_'s program for A197629

%Y Cf. A197629.

%K nonn,look

%O 1,41

%A _Jason Holland_, Oct 22 2011