A184995 Irregular triangle T, read by rows, in which row n lists the primes p <= n such that 2n-p is also prime.

2, 3, 3, 3, 5, 5, 3, 7, 3, 5, 5, 7, 3, 7, 3, 5, 11, 5, 7, 11, 3, 7, 13, 5, 11, 7, 11, 13, 3, 13, 3, 5, 11, 17, 5, 7, 13, 17, 7, 19, 3, 11, 17, 5, 11, 13, 19, 3, 7, 13, 3, 5, 17, 23, 5, 7, 11, 17, 19, 3, 7, 13, 19, 5, 11, 23, 7, 11, 13, 17, 23, 3, 13, 19, 5, 11, 17, 29, 7, 13, 17, 19, 23, 29

Offset: 2

Author

Jason Kimberley, Sep 03 2011

Keywords

• nonn
• tabf
• easy

Comments

Row n has first entry A020481(n), length A045917(n), and last entry A112823(n).

Each row is the prefix to the middle of the corresponding row of A171637.

The Goldbach conjecture states that this irregular Goldbach triangle has in each row at least one entry (A045917(n) >= 1). - Wolfdieter Lang, May 14 2016

Examples

			The irregular triangle T(n, i) starts:
n, 2*n\i  1   2   3   4   5   6 ...
2,   4    2
3,   6    3
4,   8    3
5,  10    3   5
6,  12    5
7,  14    3   7
8,  16    3   5
9,  18    5   7
10, 20    3   7
11, 22    3   5  11
12, 24    5   7  11
13, 26    3   7  13
14, 28    5  11
15, 30    7  11  13
16, 32    3  13
17, 34    3   5  11  17
18, 36    5   7  13  17
19, 38    7  19
20, 40    3  11  17
21, 42    5  11  13  19
22, 44    3   7  13
23, 46    3   5  17  23
24, 48    5   7  11  17  19
25, 50    3   7  13  19
26, 52    5  11  23
27, 54    7  11  13  17  23
28, 56    3  13  19
29, 58    5  11  17  29
30, 60    7  13  17  19  23  29
... reformatted - _Wolfdieter Lang_, May 14 2016
		

Links

• Jason Kimberley, Table of n, a(n) for n = 2..1000 (flattened 2..26552)
• OEIS (Plot 2), Plot of (n,p)
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Related triangles: A154720, A154721, A154722, A154723, A154724, A154725, A154726, A154727, A171637, A182138.

Programs

Magma

A184995 := func;
&cat[A184995(n):n in [2..30]];

Maple

T:= n-> seq(`if`(andmap(isprime, [p, 2*n-p]), p, NULL), p=2..n):
seq(T(n), n=2..40);  # Alois P. Heinz, Jan 09 2025

Mathematica

Table[Select[Prime@ Range@ PrimePi@ n, PrimeQ[2 n - #] &], {n, 2, 30}] // Flatten (* Michael De Vlieger, May 14 2016 *)
T[n_] := Table[If[PrimeQ[p] && PrimeQ[2n-p], p, Nothing], {p, 2, n}];
Table[T[n], {n, 2, 30}] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz in A182138 *)

Formula

T(n,i) = n - A182138(n,i). - Jason Kimberley, Sep 25 2012