OFFSET
2,4
COMMENTS
The Goldbach conjecture is that for any even integer 2n>=4, at least one pair of primes p and q exist such that p+q=2n. The present numbers listed here are the distances d between each prime and n, the half of the even integer 2n: d=n-p=q-n with p <= q.
See the link section for plots I added. - Jason Kimberley, Oct 04 2012
Each nonzero entry d of row n is coprime to n. For otherwise n+d would be composite. - Jason Kimberley, Oct 10 2012
LINKS
Alois P. Heinz, Rows n = 2..600, flattened
OEIS (Plot 2), Plot of (n, d)
Subplots for fixed p:
OEIS (Plot 2), A067076 vs A098090 (p=3).
OEIS (Plot 2), A089038 vs A089253 (p=5).
OEIS (Plot 2), A105760 vs A089192 (p=7).
...
OEIS (Plot 2), A153143 vs A097932 (p=19).
Wikipedia, Goldbach's conjecture
FORMULA
T(n,i) = n - A184995(n,i). - Jason Kimberley, Sep 25 2012
EXAMPLE
n=2, 2n=4, 4=2+2, p=q=2 -> d=0.
n=18, 2n=36, four prime pairs have a sum of 36: 5+31, 7+29, 13+23, 17+19, with the four distances d being 13=18-5=31-18, 11=18-7=29-18, 5=18-13=23-18, 1=18-17=19-18.
Triangle begins:
0;
0;
1;
2, 0;
1;
4, 0;
5, 3;
4, 2;
7, 3;
8, 6, 0;
MAPLE
T:= n-> seq(`if`(isprime(p) and isprime(2*n-p), n-p, NULL), p=2..n):
seq(T(n), n=2..40); # Alois P. Heinz, Apr 16 2012
PROG
(PARI) for(n=2, 18, forprime(p=2, n, if(isprime(2*n-p), print1(n-p", ")))) \\ Charles R Greathouse IV, Apr 16 2012
(Magma) A182138:= func<n|[n-p:p in PrimesUpTo(n)|IsPrime(2*n-p)]>;
&cat[A182138(n):n in [2..30]]; // Jason Kimberley, Oct 01 2012
CROSSREFS
KEYWORD
easy,nonn,tabf
AUTHOR
Jean COHEN, Apr 16 2012
STATUS
approved