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

2,1

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

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

Formula

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

Example

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

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 *)

Prog

(Magma) A184995 := func<n|[p:p in PrimesUpTo(n)|IsPrime(2*n-p)]>;
&cat[A184995(n):n in [2..30]];

Crossrefs

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

Sequence in context: A096288 A339310 A361159 * A372546 A136348 A353710

Adjacent sequences: A184992 A184993 A184994 * A184996 A184997 A184998

Keyword

nonn,tabf,easy

Author

Jason Kimberley, Sep 03 2011

Status

approved