A182138 -id:A182138 - 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

A047949 a(n) is the largest m such that n-m and n+m are both primes, or -1 if no such m exists.

Original entry on oeis.org

0, 0, 1, 2, 1, 4, 5, 4, 7, 8, 7, 10, 9, 8, 13, 14, 13, 12, 17, 16, 19, 20, 19, 22, 21, 20, 25, 24, 23, 28, 29, 28, 27, 32, 31, 34, 35, 34, 33, 38, 37, 40, 39, 38, 43, 42, 41, 30, 47, 46, 49, 50, 49, 52, 53, 52, 55, 54, 53, 48, 51, 50, 45, 62, 61, 64, 63, 62, 67, 68, 67, 66

Offset: 2

Lior Manor

A067076 is a subsequence of this sequence: when 2m+3 is prime a(m+3) = m. Moreover, it is the subsequence of records (maximal increasing subsequence): let m=a(n), with p=n-m and q=p+2m both odd primes > 3; now 3+2(m+(p-3)/2)=q and hence a(3+m+(p-3)/2) >= m+(p-3)/2 > m = a(n) but 3+m+(p-3)/2 < n. - Jason Kimberley, Aug 30 2012 and Oct 10 2012
Goldbach's conjecture says a(n) >= 0 for all n. - Robert Israel, Apr 15 2015
a(n) is the Goldbach partition of 2n which results in the maximum spread divided by 2. - Robert G. Wilson v, Jun 18 2018

			49-30=19 and 49+30=79 are primes, so a(49)=30.

T. D. Noe, Table of n, a(n) for n = 2..10000
OEIS (Plot 2), A067076 vs A098090 (n-m=3). - _Jason Kimberley_, Oct 01 2012

Cf. A047160, A067076, A182138.
Cf. A020481.
Cf. A010051.

a047949 n = if null qs then -1 else head qs  where
   qs = [m | m <- [n, n-1 .. 0], a010051' (n+m) == 1, a010051' (n-m) == 1]
-- Reinhard Zumkeller, Nov 02 2015

a:= proc(n)
local k;
  for k from n - 1 to 0 by -2 do
     if isprime(n+k) and isprime(n-k) then return(k) fi
od:
-1
end proc:
0, seq(a(n),n=3..1000); # Robert Israel, Apr 16 2015

a[2] = a[3] = 0; a[n_] := (For[m = n - 2, m >= 0, m--, If[PrimeQ[n - m] && PrimeQ[n + m], Break[]]]; m); Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Sep 04 2013 *)
lm[n_]:=Module[{m=n-2},While[!AllTrue[n+{m,-m},PrimeQ],m--];m]; Join[{0,0}, Array[ lm,70,4]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 03 2014 *)
f[n_] := Block[{q = 2}, While[q <= n && !PrimeQ[2n -q], q = NextPrime@ q]; n - q]; Array[f, 72, 2] (* Robert G. Wilson v, Jun 18 2018 *)

a(n) = {if (n==2 || n==3, return (0)); my(m = 1, lastm = -1, do = 1); while (do, if (isprime(n-m) && isprime(n+m), lastm = m); m++; if (m == n - 1, do = 0);); return (lastm);} \\ Michel Marcus, Jun 09 2013

a(n)=if(n<4,0,forprime(p=3,n-1,if(isprime(2*n-p),return(n-p)));-1) \\ Ralf Stephan, Dec 29 2013

a(n) = n - A020481(n).

a(n) = (A020482(n) - A020481(n))/2. - Gionata Neri, Apr 15 2015

Corrected by Harvey P. Dale, Dec 21 2000

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

A198292 Irregular triangle with row n being A045917(n) copies of n.

Original entry on oeis.org

2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28

Offset: 2

Jason Kimberley, Oct 03 2012

			Triangle begins:
2;
3;
4;
5, 5;
6;
7, 7;
8, 8;
9, 9;
10, 10;
11, 11, 11;

Jason Kimberley, Table of i, a(i)=T(n,k)=n for n = 2..200 (i = 2..1794)

This triangle has the same dimensions as both A182138 and A184995.

A182138:=func;
A045917:=funcA182138(n)>;
A198292:=funcA045917(n)]>;
&cat[A198292(n):n in[1..29]];

A234316 Irregular triangle T, read by rows, such that row n lists the larger parts of the Goldbach partitions of 2n (in decreasing order).

Original entry on oeis.org

2, 3, 5, 7, 5, 7, 11, 7, 13, 11, 13, 11, 17, 13, 19, 17, 11, 19, 17, 13, 23, 19, 13, 23, 17, 23, 19, 17, 29, 19, 31, 29, 23, 17, 31, 29, 23, 19, 31, 19, 37, 29, 23, 37, 31, 29, 23, 41, 37, 31, 43, 41, 29, 23, 43, 41, 37, 31, 29, 47, 43, 37, 31, 47, 41, 29, 47, 43, 41, 37, 31

Offset: 2

Wesley Ivan Hurt, Dec 23 2013

Row n has first entry A060308(n), and length A045917(n). If Goldbach's conjecture is true, then each row of the triangle contains at least 1 entry.

This is the companion irregular triangle to A184995. See the first formula. - Wolfdieter Lang, May 14 2016

			The irregular triangle T(n,i) begins:
   n | 2*n | i = 1   2   3   4   5   6 ...
  ---+-----+------------------------------
   2 |   4 |     2
   3 |   6 |     3
   4 |   8 |     5
   5 |  10 |     7   5
   6 |  12 |     7
   7 |  14 |    11   7
   8 |  16 |    13  11
   9 |  18 |    13  11
  10 |  20 |    17  13
  11 |  22 |    19  17  11
  12 |  24 |    19  17  13
  13 |  26 |    23  19  13
  14 |  28 |    23  17
  15 |  30 |    23  19  17
  16 |  32 |    29  19
  17 |  34 |    31  29  23  17
  18 |  36 |    31  29  23  19
  19 |  38 |    31  19
  20 |  40 |    37  29  23
  21 |  42 |    37  31  29  23
  22 |  44 |    41  37  31
  23 |  46 |    43  41  29  23
  24 |  48 |    43  41  37  31  29
  25 |  50 |    47  43  37  31
  26 |  52 |    47  41  29
  27 |  54 |    47  43  41  37  31
  28 |  56 |    53  43  37
  29 |  58 |    53  47  41  29
  30 |  60 |    53  47  43  41  37  31
 ... Reformatted and extended. - _Wolfdieter Lang_, May 14 2016

Eric Weisstein's World of Mathematics, Goldbach Partition
Wikipedia, Goldbach's conjecture
Index entries for sequences related to Goldbach conjecture
Index entries for sequences related to partitions

Cf. A182138, A184995.

Table[First /@ DeleteDuplicates@ Map[Sort[{#, 2 n - #}, Greater] &, Select[2 n - Prime@ Range@ PrimePi[2 n], PrimeQ]], {n, 30}] // Flatten (* Michael De Vlieger, May 15 2016 *)

for(n=2, 18, forprime(p=2, n, if(isprime(2*n-p), print1(2*n-p", ")))) \\ Ralf Stephan, Dec 26 2013

T(n,i) = 2n - A184995(n,i).

T(n,i) = n + A182138(n,i). - Ralf Stephan, Dec 26 2013

cp's OEIS Frontend

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

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A047949 a(n) is the largest m such that n-m and n+m are both primes, or -1 if no such m exists.

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

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A198292 Irregular triangle with row n being A045917(n) copies of n.

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Magma

A234316 Irregular triangle T, read by rows, such that row n lists the larger parts of the Goldbach partitions of 2n (in decreasing order).

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Mathematica

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