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).

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, 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, 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

Offset: 2

Author

Jean COHEN, Apr 16 2012

Keywords

• easy
• nonn
• look
• tabf

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

Examples

			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;
		

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

Crossrefs

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

Cf. A184995.

Programs

Magma

A182138:= func;
&cat[A182138(n):n in [2..30]]; // Jason Kimberley, Oct 01 2012

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

Mathematica

T[n_] := Table[If[PrimeQ[p] && PrimeQ[2n-p], n-p, Nothing], {p, 2, n}];
Table[T[n], {n, 2, 30}] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

PARI

for(n=2,18,forprime(p=2,n,if(isprime(2*n-p),print1(n-p", ")))) \\ Charles R Greathouse IV, Apr 16 2012

Formula

T(n,i) = n - A184995(n,i). - Jason Kimberley, Sep 25 2012