login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 (list; graph; refs; listen; history; text; internal format)
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: 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 17 06:30 EDT 2021. Contains 343965 sequences. (Running on oeis4.)