A182138 - OEIS

A182138

Irregular triangle T, read by rows, in which row n lists the distances between n and the two primes whose sum makes 2n in decreasing order (Goldbach conjecture).

%I #54 Sep 08 2022 08:45:54

%S 0,0,1,2,0,1,4,0,5,3,4,2,7,3,8,6,0,7,5,1,10,6,0,9,3,8,4,2,13,3,14,12,

%T 6,0,13,11,5,1,12,0,17,9,3,16,10,8,2,19,15,9,20,18,6,0,19,17,13,7,5,

%U 22,18,12,6,21,15,3,20,16,14,10,4,25,15,9,24,18,12,0,23,17,13,11,7,1

%N Irregular triangle T, read by rows, in which row n lists the distances between n and the two primes whose sum makes 2n in decreasing order (Goldbach conjecture).

%C 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.

%C See the link section for plots I added. - _Jason Kimberley_, Oct 04 2012

%C Each nonzero entry d of row n is coprime to n. For otherwise n+d would be composite. - _Jason Kimberley_, Oct 10 2012

%H Alois P. Heinz, <a href="/A182138/b182138.txt">Rows n = 2..600, flattened</a>

%H OEIS (Plot 2), <a href="/plot2a?name1=A198292&name2=A182138&tform1=untransformed&tform2=untransformed&shift=0&radiop1=xy&drawpoints=true">Plot of (n, d) </a>

%H Subplots for fixed p:

%H OEIS (Plot 2), <a href="/plot2a?name1=A067076&name2=A098090&tform1=untransformed&tform2=untransformed&shift=0&radiop1=xy&drawpoints=true">A067076 vs A098090</a> (p=3).

%H OEIS (Plot 2), <a href="/plot2a?name1=A089038&name2=A089253&tform1=untransformed&tform2=untransformed&shift=1&radiop1=xy&drawpoints=true">A089038 vs A089253</a> (p=5).

%H OEIS (Plot 2), <a href="/plot2a?name1=A105760&name2=A089192&tform1=untransformed&tform2=untransformed&shift=2&radiop1=xy&drawpoints=true">A105760 vs A089192</a> (p=7).

%H ...

%H OEIS (Plot 2), <a href="/plot2a?name1=A153143&name2=A097932&tform1=untransformed&tform2=untransformed&shift=6&radiop1=xy&drawpoints=true">A153143 vs A097932</a> (p=19).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>

%F T(n,i) = n - A184995(n,i). - _Jason Kimberley_, Sep 25 2012

%e n=2, 2n=4, 4=2+2, p=q=2 -> d=0.

%e 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.

%e Triangle begins:

%e 0;

%e 1;

%e 2, 0;

%e 1;

%e 4, 0;

%e 5, 3;

%e 4, 2;

%e 7, 3;

%e 8, 6, 0;

%p T:= n-> seq(`if`(isprime(p) and isprime(2*n-p), n-p, NULL), p=2..n):

%p seq(T(n), n=2..40); # _Alois P. Heinz_, Apr 16 2012

%o (PARI) for(n=2,18,forprime(p=2,n,if(isprime(2*n-p),print1(n-p", ")))) \\ _Charles R Greathouse IV_, Apr 16 2012

%o (Magma) A182138:= func<n|[n-p:p in PrimesUpTo(n)|IsPrime(2*n-p)]>;

%o &cat[A182138(n):n in [2..30]]; // _Jason Kimberley_, Oct 01 2012

%Y Cf. A045917 (row lengths), A047949 (first column), A047160 (last elements of rows).

%Y Cf. A184995.

%K easy,nonn,tabf

%O 2,4

%A _Jean COHEN_, Apr 16 2012