

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).


6



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
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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=np=qn 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=185=3118, 11=187=2918, 5=1813=2318, 1=1817=1918.
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*np), np, 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*np), print1(np", ")))) \\ Charles R Greathouse IV, Apr 16 2012
(MAGMA) A182138:= func<n[np:p in PrimesUpTo(n)IsPrime(2*np)]>;
&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: A273821 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



