A198292 -id:A198292 - cp's OEIS frontend

A045917 From Goldbach problem: number of decompositions of 2n into unordered sums of two primes.

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3, 7, 9, 6, 5, 8, 7, 8, 11, 6, 5, 12, 4, 8, 11, 5, 8, 10, 5, 6, 13, 9, 6, 11, 7, 7, 14, 6, 8, 13, 5, 8, 11, 7, 9

Offset: 1

Felice Russo

Note that A002375 (which differs only at the n = 2 term) is the main entry for this sequence.
The graph of this sequence is called Goldbach's comet. - David W. Wilson, Mar 19 2012
This is the row length sequence of A182138, A184995 and A198292. - Jason Kimberley, Oct 03 2012
The Goldbach conjecture states that a(n) > 0 for n >= 2. - Wolfdieter Lang, May 14 2016
With the second Maple program, the command G(2n) yields all the unordered pairs of prime numbers having sum 2n; caveat: a pair {a,a} is listed as {a}. Example: G(26) yields {{13}, {3,23}, {7,19}}. The command G(100000) yields 810 pairs very fast. - Emeric Deutsch, Jan 03 2017
Conjecture: Let p denote any prime in any decomposition of 2n. 4 and 6 are the only numbers n such that 2n + p is prime for every p. - Ivan N. Ianakiev, Apr 06 2017
Conjecture: For all m >= 0, there exists at least one possible value of n such that a(n) = m. - Ahmad J. Masad, Jan 06 2018
The previous conjecture is related to the sequence A053033. - Ahmad J. Masad, Dec 09 2019
Conjecture: For each k >= 0, there exists a minimum sufficiently large number r that depends on k such that for each n >= r, a(n) > k. - Ahmad J. Masad, Jan 08 2020
Conjecture: If the previous conjecture is true, then for each m >= 0, the number of terms that are equal to (m+1) is larger than the number of terms that are equal to m. - Ahmad J. Masad, Jan 08 2020
Also, the number of equidistant prime pairs in Goldbach's Prime Triangle for integers n > 2. An equidistant prime pair is a pair of not necessarily different prime numbers (p1, p2) that have the same distance d >= 0 from an integer n, i.e., so that p1 = n - d and p2 = n + d. - Jörg Winkelmann, Mar 05 2025

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, pages 236-257.
H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.

H. J. Smith, Table of n, a(n) for n = 1..20000
M. Herkommer, Goldbach Conjecture Research.
Eric Weisstein's World of Mathematics, Goldbach Partition.
Wikipedia, Goldbach's conjecture.
G. Xiao, WIMS server, Goldbach
Index entries for sequences related to Goldbach conjecture
J. Winkelmann, Goldbach's Prime Triangle — A Recreational Math Journey with an Introduction to Equidistant Primes.

Cf. A002375 (the main entry for this sequence (which differs only at the n=2 term)).
Cf. A023036 (first appearance of n), A000954 (last (assumed) appearance of n).
Cf. also A000040, A182138, A184995, A185297, A187129, A010051, A198292, A035026, A224708, A224709.
Cf. A064911, A105020.

a045917 n = sum $ map (a010051 . (2 * n -)) $ takeWhile (<= n) a000040_list
-- Reinhard Zumkeller, Sep 02 2013

[#RestrictedPartitions(2*n,2,Set(PrimesInInterval(1,2*n))):n in [1..100]]; // Marius A. Burtea, Jan 23 2020

A045917 := proc(n)
    local a,i ;
    a := 0 ;
    for i from 1 to n do
        if isprime(i) and isprime(2*n-i) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 01 2013
# second Maple program:
G := proc (n) local g, j: g := {}: for j from 2 to (1/2)*n do if isprime(j) and isprime(n-j) then g := `union`(g, {{n-j, j}}) end if end do: g end proc: seq(nops(G(2*n)), n = 1 .. 98); # Emeric Deutsch, Jan 03 2017

f[n_] := Length[Select[2n - Prime[Range[PrimePi[n]]], PrimeQ]]; Table[ f[n], {n, 100}] (* Paul Abbott, Jan 11 2005 *)
nn = 10^2; ps = Boole[PrimeQ[Range[1,2*nn,2]]]; Join[{0,1}, Table[Sum[ps[[i]] ps[[n-i+1]], {i, Ceiling[n/2]}], {n, 3, nn}]] (* T. D. Noe, Apr 13 2011 *)

a(n)=my(s);forprime(p=2,n,s+=isprime(2*n-p));s \\ Charles R Greathouse IV, Mar 27 2012

from sympy import isprime
def A045917(n):
    x = 0
    for i in range(2,n+1):
        if isprime(i) and isprime(2*n-i):
            x += 1
    return x # Chai Wah Wu, Feb 24 2015

From Halberstam and Richert: a(n) < (8+0(1))*c(n)*n/log(n)^2 where c(n) = Product_{p>2} (1 - 1/(p-1)^2)*Product_{p|n, p>2} (p-1)/(p-2). It is conjectured that the factor 8 can be replaced by 2. - Benoit Cloitre, May 16 2002
a(n) = ceiling(A035026(n) / 2) = (A035026(n) + A010051(n)) / 2.
a(n) = Sum_{i=2..n} floor(2/Omega(i*(2*n-i))). - Wesley Ivan Hurt, Jan 24 2013
a(n) = A224709(n) + (primepi(2n-2) - primepi(n-1)) + primepi(n) + 1 - n. - Anthony Browne, May 03 2016
a(n) = A224708(2n) - A224708(2n+1) + A010051(n). - Anthony Browne, Jun 26 2016
a(n) = Sum_{k=n*(n-1)/2+2..n*(n+1)/2} A064911(A105020(k-1)). - Wesley Ivan Hurt, Sep 11 2021
a(n) = omega(A362641(n)) = omega(A362640(n)). - Wesley Ivan Hurt, Apr 28 2023

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).

Original entry on oeis.org

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

Jean COHEN, Apr 16 2012

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

			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;

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

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

Cf. A184995.

A182138:= func;
&cat[A182138(n):n in [2..30]]; // Jason Kimberley, Oct 01 2012

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

T[n_] := Table[If[PrimeQ[p] && PrimeQ[2n-p], n-p, Nothing], {p, 2, n}];
Table[T[n], {n, 2, 30}] // Flatten (* Jean-François Alcover, Jan 09 2025, after Alois P. Heinz *)

for(n=2,18,forprime(p=2,n,if(isprime(2*n-p),print1(n-p", ")))) \\ Charles R Greathouse IV, Apr 16 2012

T(n,i) = n - A184995(n,i). - Jason Kimberley, Sep 25 2012

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

Original entry on oeis.org

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

Jason Kimberley, Sep 03 2011

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

			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

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

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

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

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

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 *)

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

cp's OEIS Frontend

A045917 From Goldbach problem: number of decompositions of 2n into unordered sums of two primes.

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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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula