A197629 -id:A197629 - cp's OEIS frontend

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

Jason Holland, Oct 22 2011

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

			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)

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

Cf. A197629.

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

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

A197640 Numbers not representable as the sum of two coprime, squarefree, composite, positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 58, 60, 62, 63, 64, 66, 70, 72, 75, 78, 80, 82, 84, 88, 90, 96, 100, 102, 105, 108, 110, 114, 120, 126, 130, 132, 138, 140, 144, 150, 156, 168, 180, 190, 204, 210, 240, 270, 330, 420

Offset: 1

Jason Holland, Oct 16 2011

nonn

a(87) = 420 is probably the last term.

Cf. A197629, A096076.

is(n)=for(k=6,n\2,if(gcd(k,n-k)==1&&!isprime(k)&&!isprime(n-k)&&issquarefree(k)&&issquarefree(n-k),return(0)));1 \\ Charles R Greathouse IV, Oct 18 2011

cp's OEIS Frontend

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

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