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

2,4

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

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

Cf. A184995.

Sequence in context: A363519 A108643 A133838 * A258123 A121583 A228924

Adjacent sequences: A182135 A182136 A182137 * A182139 A182140 A182141

Keyword

easy,nonn,tabf

Author

Jean COHEN, Apr 16 2012

Status

approved