A198255 Number of ways to write n as the sum of two coprime squarefree semiprimes.

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, 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, 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

Offset

1,41

Comments

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.

There are interesting bands in the scatterplot of this sequence. - Antti Karttunen, Sep 23 2018

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Example

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).
From Antti Karttunen, Sep 23 2018: (Start)
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.
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.
(End)

Prog

(MATLAB)
function [asubn] = sps(n)
% Returns the number of coprime, squarefree, semiprime, partitions a+b of n.
r = 0; % r is the number of sps's of n
k=6;
while k < n/2,
    if gcd(k, n-k)==1
         if length(factor(k)) == 2
             if length(factor(n-k)) == 2
                 if prod(diff(factor(k)))*prod(diff(factor(n-k))) > 0
                    r = r + 1;
                 end
             end
         end
    end
    k = k + 1;
end
asubn = r;
end
(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

Crossrefs

Cf. A197629.

Sequence in context: A318982 A069851 A197629 * A290542 A359300 A029834

Adjacent sequences: A198252 A198253 A198254 * A198256 A198257 A198258

Keyword

nonn,look

Author

Jason Holland, Oct 22 2011

Status

approved