A045917 -id:A045917 - cp's OEIS frontend

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

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]];

A045919 Partial sum of Goldbach numbers A045917.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 10, 12, 14, 17, 20, 23, 25, 28, 30, 34, 38, 40, 43, 47, 50, 54, 59, 63, 66, 71, 74, 78, 84, 87, 92, 98, 100, 105, 111, 116, 121, 128, 132, 137, 145, 150, 154, 163, 167, 172

Offset: 1

Felice Russo

If the Goldbach conjecture is false then a(n)=a(n+1) for some n.

C. Clawson, Mathematical mysteries, Plenum Press 1996, p. 241.

Vincenzo Librandi, Table of n, a(n) for n = 1..300
Index entries for sequences related to Goldbach conjecture

Cf. A002375, A045917, A045922.

Table[Sum[Sum[(PrimePi[2 k - i] - PrimePi[2 k - i - 1]) (PrimePi[i] - PrimePi[i - 1]), {i, k}], {k, n}], {n, 100}] (* Wesley Ivan Hurt, Apr 07 2018 *)

a(n) = Sum_{k=1..n} Sum_{i=1..k} A010051(i) * A010051(2k-i). - Wesley Ivan Hurt, Apr 07 2018

A254713 All numbers k such that the number of distinct parts of all A045917(k) Goldbach partitions of 2k is prime.

Original entry on oeis.org

4, 5, 6, 7, 11, 13, 17, 19, 23, 29, 31, 53, 59, 61, 67, 73, 83, 89, 97, 101, 103, 109, 113, 127, 131, 139, 151, 157, 163, 173, 179, 191, 193, 199, 223, 227, 229, 251, 263, 271, 307, 313, 337, 347, 353, 359, 367, 379, 389, 401, 449, 479, 521, 523, 577, 587, 599, 601, 607, 613, 631, 643

Offset: 1

Ivan N. Ianakiev, Feb 06 2015

Conjecture: a(k) is prime for k > 3. Tested for k up to 3*10^4.

			For k = 4, 2k = 8. The number of the distinct Goldbach parts of 8 (3 and 5) is prime, therefore 4 is in the sequence.
5 is in the sequence because the 2 = A045917(5) Goldbach partitions of 10 are 5 + 5 and 3 + 7, and there are 3 distinct parts, namely 3, 5 and 7. - _Wolfdieter Lang_, Feb 23 2015

Eric Weisstein's World of Mathematics, Goldbach Partition.

Cf. A035026, A002372, A045917.

lstIn={};lstFin={};
goldPart[x_]:=Module[{h=x/2},While[h>1,If[And[PrimeQ[h],PrimeQ[x-h]],AppendTo[lstIn,{h,x-h}]];h--];
lstFin=Length[Union[Flatten[lstIn]]];lstIn={};lstFin];
a254713=Flatten[Position[PrimeQ[goldPart/@Range[2,2002,2]],True]]

Edited. Wolfdieter Lang, Feb 23 2015

A343185 Numbers k such that 2*k is a multiple of A045917(k).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 19, 30, 33, 34, 35, 36, 40, 44, 45, 46, 56, 60, 70, 76, 80, 92, 100, 112, 114, 128, 130, 140, 145, 148, 182, 184, 209, 210, 221, 228, 238, 247, 270, 276, 286, 297, 324, 344, 372, 399, 408, 410, 425, 429, 437, 444, 460, 468, 475, 504, 506, 507, 510

Offset: 1

J. M. Bergot and Robert Israel, Apr 18 2021

nonn

			a(4) = 5 is a term because 2*5 = 10 is a multiple of A045917(5) = 2.

Robert Israel, Table of n, a(n) for n = 1..600

Cf. A045917.

P:= select(isprime, [2,seq(i,i=3..2000,2)]):
filter:= proc(n) local k1, k2;
   k1:= ListTools:-BinaryPlace(P,n+1);
   k2:= ListTools:-BinaryPlace(P,2*n+1);
   2*n mod nops(convert(P[1..k1],set) intersect map(t -> 2*n-t, convert(P[k1..k2],set))) = 0
end proc:
select(filter, [$2..1000]);

A350455 T(n,k) is the k-th semiprime whose sum of prime factors equals 2n, triangle T(n,k), n>=2, 1<=k<=A045917(n), read by rows.

Original entry on oeis.org

4, 9, 15, 21, 25, 35, 33, 49, 39, 55, 65, 77, 51, 91, 57, 85, 121, 95, 119, 143, 69, 133, 169, 115, 187, 161, 209, 221, 87, 247, 93, 145, 253, 289, 155, 203, 299, 323, 217, 361, 111, 319, 391, 185, 341, 377, 437, 123, 259, 403, 129, 205, 493, 529, 215, 287, 407

Offset: 2

Alois P. Heinz, Dec 31 2021

Assuming Goldbach's conjecture, no row is empty.

			Triangle T(n,k) begins:
    4;
    9;
   15;
   21,  25;
   35     ;
   33,  49;
   39,  55;
   65,  77;
   51,  91;
   57,  85, 121;
   95, 119, 143;
   69, 133, 169;
  115, 187     ;
  161, 209, 221;
   87, 247     ;
   93, 145, 253, 289;
  155, 203, 299, 323;
  ...

Alois P. Heinz, Rows n = 2..1000, flattened
Wikipedia, Goldbach's conjecture

Column k=1 gives A073046.
Last elements of rows give A102084.
Row sums give A228553.
Row products give A337568.
Row lengths give A045917.
Cf. A000040, A001358, A046315, A350419.

T:= n-> seq(`if`(andmap(isprime, [h, 2*n-h]), h*(2*n-h), [][]), h=2..n):
seq(T(n), n=2..30);

A239282 a(n) = A045917(n)*prime(n).

Original entry on oeis.org

0, 3, 5, 7, 22, 13, 34, 38, 46, 58, 93, 111, 123, 86, 141, 106, 236, 244, 134, 213, 292, 237, 332, 445, 388, 303, 515, 321, 436, 678, 381, 655, 822, 278, 745, 906, 785, 815, 1169, 692, 895, 1448, 955, 772, 1773, 796, 1055, 1561, 681, 1374, 1864, 1195, 1446, 2008

Offset: 1

Raghavendra Ugare, Mar 14 2014

nonn

Cf. A045917, A000040

(*Returns the various ways a number (presumed to be even) can be split as a sum of 2 Primes.*)
getGoldBachSplits[n_Integer] := Module[{i, splits = {}, a, b},
  (
   For[i = 1,
    Prime[i] < n,
    i++,
    a = Prime[i] ;
    b = n - Prime[i];
    If[PrimeQ[b],
     If[MemberQ[splits, {b, a}], Null,
      AppendTo[splits, {a, b}]],
     Null];
    ]; (*End For-loop...*)
   splits
   )
  ]
(* Now we generate our series...*)
series[n_] :=
Module[{}, (Table[Prime[i]*Length[getGoldBachSplits[2 i]], {i, 2, n}])]

A245629 Numbers n such that A000203(2n) divides 2n*A045917(n).

Original entry on oeis.org

1, 14, 42, 60, 336, 1638, 2160, 4064, 4130, 4464, 5148, 6678, 7900, 9856, 12192, 13144, 16464, 23220, 24206, 26001, 28665, 44460, 49680, 53464, 105656, 117800, 125685, 158160, 159489, 168597, 173060, 232128, 276080, 309504, 320580, 372384, 475488, 542430, 580072, 613500, 699112, 708900, 787644, 834561, 843200, 885456, 914872, 1215396

Offset: 1

Ivan N. Ianakiev, Jul 27 2014

nonn

Conjecture: 14 is the only natural number n for which A000203(2*n) equals 2*n*A045917(n).

Conjecture above is confirmed for n < 10^5. - Derek Orr, Jul 27 2014

			A000203(2*14) = 56, which divides 2*14*A045917(14), which is also 56. So 14 is a member of this sequence.

Cf. A000203, A045917.

f[n_] := Length@ Select[ 2n - Prime@ Range@ PrimePi@ n, PrimeQ]; fQ[n_] := Mod[ 2n*f[n], DivisorSigma[1, 2n]] == 0; k = 1; lst = {}; While[k < 1250001, If[ fQ@ k, AppendTo[lst, k]; Print@ k]; k++]; lst (* Robert G. Wilson v, Aug 07 2014 *)

for(n=1,10^7,my(s);forprime(p=2,n,s+=isprime(2*n-p));d=divisors(2*n);if(2*n*s%(sum(i=1,#d,d[i]))==0,print1(n,", "))) \\ Derek Orr, Jul 27 2014

a(18)-a(24) from Derek Orr, Jul 27 2014

a(25)-a(48) from Robert G. Wilson v, Aug 07 2014

A002375 From Goldbach conjecture: number of decompositions of 2n into an unordered sum of two odd primes.

Original entry on oeis.org

0, 0, 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

N. J. A. Sloane

A weaker form of this conjecture, the ternary form, was proved by Helfgott (see link below). - T. D. Noe, May 14 2013
The Goldbach conjecture is that for n >= 3, this sequence is always positive.
This has been checked up to at least 10^18 (see A002372).
With the exception of the n=2 term, identical to A045917.
The conjecture has been verified up to 3 * 10^17 (see MathWorld link). - Dmitry Kamenetsky, Oct 17 2008
Languasco and Zaccagnini proved that, where Lambda is the von Mangoldt function, and R(n) = Sum_{i + j = n} Lambda(i)*Lambda(j) is the counting function for the Goldbach numbers, and for N >= 2 and assume the Riemann hypothesis (RH) holds, then Sum_{n = 1..N} R(n) = (N^2)/2 - 2*Sum_{rho} ((N^(rho+1))/(rho*(rho+1))) + O(N * log^3 N).
If 2n is the sum of two distinct primes, then neither prime divides 2n. - Christopher Heiling, Feb 28 2017

			2 and 4 are not the sum of 2 odd primes, so a(1) = a(2) = 0; 6 = 3 + 3 (one way, so a(3) = 1); 8 = 3 + 5 (so a(4) = 1); 10 = 3 + 7 = 5 + 5 (so a(5) = 2); etc.

Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, Pages 236-257.
Apostolos K. Doxiadis, Uncle Petros and Goldbach's Conjecture, Bloomsbury Pub. PLC USA, 2000.
D. A. Grave, Traktat z Algebrichnogo Analizu (Monograph on Algebraic Analysis). Vol. 2, p. 19. Vidavnitstvo Akademiia Nauk, Kiev, 1938.
H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 80.
N. V. Maslova, On the coincidence of Grünberg-Kegel graphs of a finite simple group and its proper subgroup, Proceedings of the Steklov Institute of Mathematics April 2015, Volume 288, Supplement 1, pp 129-141; Original Russian Text: Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2014, Vol. 20, No. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. J. Smith, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
J.-M. Deshouillers, H. J. J. te Riele and Y. Saouter, New Experimental Results Concerning the Goldbach Conjecture, Modelling, Analysis and Simulation [MAS], R 9804, p.1-12, Technical report, 1998.
James A. Farrugia, Brun's 1920 Theorem on Goldbach's Conjecture, Master's Thesis, Utah State University, All Graduate Theses and Dissertations (2018). 7153.
H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252 [math.NT], 2012-2013.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013-2014.
H. A. Helfgott, The ternary Goldbach conjecture is true, arxiv:1312.7748 [math.NT], 2013.
H. A. Helfgott, The ternary Goldbach problem, arXiv:1404.2224 [math.NT], 2014.
M. Herkommer, Goldbach Conjecture Research
A. V. Kumchev and D. I. Tolev, An invitation to additive number theory, arXiv:math/0412220 [math.NT], 2004.
Alessandro Languasco and Alessandro Zaccagnini, The number of Goldbach representations of an integer, Proc. Amer. Math. Soc. 140 (2012), 795-804 (preliminary version, arXiv:1011.3198 [math.NT], 2010).
Jörg Richstein, Verifying the Goldbach conjecture up to 4 * 10^14, Math. Comput., 70 (2001), 1745-1749.
Vladimir Shevelev, Binary additive problems: recursions for numbers of representations, arXiv:0901.3102 [math.NT], 2009-2013.
Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Math. Comp. 61 (1993), pp. 931-934.
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

See also A061358. Cf. A002372 (ordered sums), A002373, A002374, A045917.
A023036 is (essentially) the first appearance of n and A000954 is the last (assumed) appearance of n.
Cf. A065091, A010051, A001031 (a weaker form of the conjecture).

a002375 n = sum $ map (a010051 . (2 * n -)) $ takeWhile (<= n) a065091_list
-- Reinhard Zumkeller, Sep 02 2013

A002375 := func; [A002375(n):n in[1..98]];

A002375 := proc(n) local s, p; s := 0; p := 3; while p<2*n do s := s+x^p; p := nextprime(p) od; (coeff(s^2, x, 2*n)+coeff(s,x,n))/2 end; [seq(A002375(n), n=1..100)];
a:=proc(n) local c,k; c:=0: for k from 1 to floor((n-1)/2) do if isprime(2*k+1)=true and isprime(2*n-2*k-1)=true then c:=c+1 else c:=c fi od end: A:=[0,0,seq(a(n),n=3..98)]; # Emeric Deutsch, Aug 27 2007
g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=2..j),j=2..50): seq(coeff(g,x,2*n), n =1..98); # Emeric Deutsch, Aug 27 2007

f[n_] := Length[ Select[2n - Prime[ Range[2, PrimePi[n]]], PrimeQ]]; Table[ f[n], {n, 100}] (* Paul Abbott, Jan 11 2005 *)
nn = 10^2; ps = Boole[PrimeQ[Range[1,2*nn,2]]]; Table[Sum[ps[[i]] ps[[n-i+1]], {i, Ceiling[n/2]}], {n, nn}] (* T. D. Noe, Apr 13 2011 *)
Table[Count[IntegerPartitions[2n,{2}],?(AllTrue[#,PrimeQ]&&FreeQ[#,2]&)],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Mar 01 2018 *)
j[n_] := If[PrimeQ[2 n - 1], 2 n - 1, 0]; A085090 = Array[j, 98];
r[n_] := Table[A085090[[k]] + A085090[[n - k + 1]], {k, 1, n}];
countzeros[l_List] := Sum[KroneckerDelta[0, k], {k, l}];
Table[((x = n - 2 countzeros[A085090[[1 ;; n]]] + countzeros[r[n]]) +
KroneckerDelta[OddQ[x], True])/2, {n, 1, 98}] (* Fred Daniel Kline, Aug 30 2018 *)

A002375 := proc(n) local s,p; begin s := 0; p := 3; repeat if isprime(2*n-p) then s := s+1 end_if; p := nextprime(p+2); until p>n end_repeat; s end_proc:

A002375(n)=sum(i=2,primepi(n),isprime(2*n-prime(i))) /* ...i=1... gives A045917 */

apply( {A002375(n,s=0,N=2*n)=forprime(p=n, N-3, isprime(N-p)&&s++);s}, [1..100]) \\ M. F. Hasler, Jan 03 2023

from sympy import primerange, isprime
def A002375(n): return sum(1 for p in primerange(3,n+1) if isprime((n<<1)-p)) # Chai Wah Wu, Feb 20 2025

def A002375(n):
    P = primes(3, n+1)
    M = (2*n - p for p in P)
    F = [k for k in M if is_prime(k)]
    return len(F)
[A002375(n) for n in (1..98)] # Peter Luschny, May 19 2013

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. Is a(n) > n/log(n)^2 for n large enough? - Benoit Cloitre, May 20 2002
a(n) = ceiling(A002372(n)/2). - Emeric Deutsch, Jul 14 2004
G.f.: Sum_{j>=2} Sum_{i=2..j} x^(p(i) + p(j)), where p(k) is the k-th prime. - Emeric Deutsch, Aug 27 2007
Not very efficient: a(n) = (Sum_{i=1..n} (pi(i) - pi(i-1)) * (pi(2n-i) - pi(2n-i-1))) - floor(2/n)*floor(n/2). - Wesley Ivan Hurt, Jan 06 2013
For n >= 2, a(n) = Sum_{3 <= p <= n, p is prime} A(2*n - p) - binomial(A(n), 2) - a(n-1) - a(n-2) - ... - a(1), where A(n) = A033270(n) (see Example 1 in link of V. Shevelev). - Vladimir Shevelev, Jul 08 2013

Beginning corrected by Paul Zimmermann, Mar 15 1996
More terms from James Sellers
Edited by Charles R Greathouse IV, Apr 20 2010

A061358 Number of ways of writing n = p+q with p, q primes and p >= q.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 3, 0, 3, 1, 3, 0, 2, 0, 3, 1, 2, 1, 4, 0, 4, 0, 2, 1, 3, 0, 4, 1, 3, 1, 4, 0, 5, 1, 4, 0, 3, 0, 5, 1, 3, 0, 4, 0, 6, 1, 3, 1, 5, 0, 6, 0, 2, 1, 5, 0, 6, 1, 5, 1, 5, 0, 7, 0, 4, 1, 5, 0, 8, 1, 5, 0, 4, 0, 9, 1, 4, 0, 5, 0, 7, 0, 3, 1, 6, 0, 8, 1, 5, 1

Offset: 0

Amarnath Murthy, Apr 28 2001

For an odd number n, a(n) = 0 if n-2 is not a prime, otherwise a(n) = 1.
For n > 1, a(2n) is at least 1, according to Goldbach's conjecture.
a(A014092(n)) = 0; a(A014091(n)) > 0; a(A067187(n)) = 1. - Reinhard Zumkeller, Nov 22 2004
Number of partitions of n into two primes.
Number of unordered ways of writing n as the sum of two primes.
a(2*n) = A068307(2*n+2). - Reinhard Zumkeller, Aug 08 2009
4*a(n) is the total number of divisors of all primes p and q such that n = p+q and p >= q. - Wesley Ivan Hurt, Mar 05 2016
Indices where a(n) = 0 correspond to A164376 UNION A025584. - Bill McEachen, Jan 31 2024

			a(22) = 3 because 22 can be written as 3+19, 5+17 and 11+11.

a(2n) is A045917.
Cf. A067187, A067188, A067189, A067190, A067191, A063610, A073610, A107318.
Column k=2 of A117278.

[#RestrictedPartitions(n,2,{p:p in PrimesUpTo(1000)}):n in [0..100] ] // Marius A. Burtea, Jan 19 2019

g:=sum(sum(x^(ithprime(i)+ithprime(j)),i=1..j),j=1..30): gser:=series(g,x=0,110): seq(coeff(gser,x,n),n=0..105); # Emeric Deutsch, Apr 03 2006

a[n_] := Length[Select[n - Prime[Range[PrimePi[n/2]]], PrimeQ]]; Table[a[n], {n, 0, 100}] (* Paul Abbott, Jan 11 2005 *)
With[{nn=110},CoefficientList[Series[Sum[x^(Prime[i]+Prime[j]),{j,nn},{i,j}],{x,0,nn}],x]] (* Harvey P. Dale, Aug 17 2017 *)
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,PrimeQ]&)],{n,0,110}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 03 2021 *)

a(n)=my(s);forprime(q=2,n\2,s+=isprime(n-q));s \\ Charles R Greathouse IV, Mar 21 2013

from sympy import primerange, isprime, floor
def a(n):
    s=0
    for q in primerange(2, n//2 + 1): s+=isprime(n - q)
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 30 2017

G.f.: Sum_{j>0} Sum_{i=1..j} x^(p(i)+p(j)), where p(k) is the k-th prime. - Emeric Deutsch, Apr 03 2006
A065577(n) = a(10^n).
From Wesley Ivan Hurt, Jan 04 2013: (Start)
a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A010051(n-i).
a(n) = Sum_{i=1..floor(n/2)} floor((A010051(i) + A010051(n-i))/2). (End)
a(n) + A062610(n) + A062602(n) = A004526(n). - R. J. Mathar, Sep 10 2021
a(n) = Sum_{k=floor((n-1)^2/4)+1..floor(n^2/4)} c(A339399(2k-1)) * c(A339399(2k)), where c = A010051. - Wesley Ivan Hurt, Jan 19 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

Comments edited by Zak Seidov, May 28 2014

A002372 Goldbach conjecture: number of decompositions of 2n into ordered sums of two odd primes.

Original entry on oeis.org

0, 0, 1, 2, 3, 2, 3, 4, 4, 4, 5, 6, 5, 4, 6, 4, 7, 8, 3, 6, 8, 6, 7, 10, 8, 6, 10, 6, 7, 12, 5, 10, 12, 4, 10, 12, 9, 10, 14, 8, 9, 16, 9, 8, 18, 8, 9, 14, 6, 12, 16, 10, 11, 16, 12, 14, 20, 12, 11, 24, 7, 10, 20, 6, 14, 18, 11, 10, 16, 14, 15, 22, 11, 10, 24, 8, 16, 22, 9, 16, 20, 10

Offset: 1

N. J. A. Sloane

The weak form of this conjecture was proved by Helfgott (see link below). - T. D. Noe, May 14 2013
Goldbach conjectured in 1742 that for n >= 3, this sequence never vanishes. This is still unproved.
Number of different primes occurring when 2n is expressed as p1+q1 = ... = pk+qk where pk,qk are odd primes with pk <= qk. For example when n=5: 10 = 3+7 = 5+5, we can see 3 different primes so a(5) = 3. - Naohiro Nomoto, Feb 24 2002
Comments from Tomás Oliveira e Silva to Number Theory List, Feb 05 2005: With the help of Siegfied "Zig" Herzog of PSU, I was able to verify the Goldbach conjecture up to 2e17. Let 2n=p+q, with p and q prime be a Goldbach partition of 2n. In a minimal Goldbach partition p is as small as possible. The largest p of a minimal Goldbach partition found was 8443 and is needed for 2n=121005022304007026. Furthermore, the largest prime gap found was 1220-1; it occurs after the prime 80873624627234849.
Comments from Tomás Oliveira e Silva to Number Theory List, Apr 26 2007: With the help of Siegfried "Zig" Herzog, the NCSA and others, I have just finished the verification of the Goldbach conjecture up to 1e18. This took about 320 years of CPU time, including a double-check of the results up to 1e17. As expected, no counterexample to the conjecture was found. As side results, the number of twin primes up to 1e18 was also computed, as was the number of primes in each of the residue classes modulo 120. Also, the number of occurrences of each (observed) prime gap was also recorded.
For n > 2 we have a(n) = 2*A002375(n)-1 if n is prime and a(n) = 2*A002375(n) if n is composite. - Emeric Deutsch, Jul 14 2004
For n > 2, a(n) = 2*A002375(n) - A010051(n). - Jason Kimberley, Aug 31 2011
a(n) = Sum_{p odd prime < 2*n} A010051(2*n - p). - Reinhard Zumkeller, Oct 19 2011
There is an interesting similarity with square numbers: The number of divisors of n is odd iff n is square (A000290).  The number of decompositions of 2n into ordered sums of two primes (equaling the number of the unique primes in all such decompositions) is odd iff n is prime. - Ivan N. Ianakiev, Feb 28 2015

			2 has no such decompositions, so a(1) = 0.
Idem for 4, whence a(2) = 0.
6 = 3+3, so a(3) = 1.
8 = 3+5 = 5+3, so a(4) = 2.
10 = 5+5 = 3+7 = 7+3, so a(5) = 3.
12 = 5+7 = 7+5; so a(6) = 2, etc.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 9.
R. K. Guy, Unsolved problems in number theory, second edition, Springer-Verlag, 1994.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Section 2.8 (for Goldbach conjecture).
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 79, 80.
N. Pipping, Neue Tafeln für das Goldbachsche Gesetz nebst Berichtigungen zu den Haussnerschen Tafeln, Finska Vetenskaps-Societeten, Comment. Physico Math. 4 (No. 4, 1927), pp. 1-27.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. L. Stein and P. R. Stein, Tables of the Number of Binary Decompositions of All Even Numbers Less Than 200,000 into Prime Numbers and Lucky Numbers. Report LA-3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.

T. D. Noe, Table of n, a(n) for n = 1..10000
Peter B. Borwein, Stephen K. K. Choi, Greg Martin, Charles L. Samuels, Polynomials whose reducibility is related to the Goldbach conjecture, arXiv:1408.4881 [math.NT], 2014 (see R(N) on page 1).
J. -M. Deshouillers, H. J. J. te Riele, Y. Saouter, New experimental results concerning the Goldbach conjecture, preprint, Centrum Wiskunde & Informatica, 1998.
J. -M. Deshouillers, H. J. J. te Riele, Y. Saouter, New experimental results concerning the Goldbach conjecture, Algorithmic number theory (Portland, OR, 1998), 204-215, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998.
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 2013-2014.
Yan Kun, Li Hou Biao, Divisor Goldbach Conjecture and its Partition Number, arXiv:1603.05233 [math.NT], 2016.
T. Oliveira e Silva, Goldbach conjecture verification.
T. Oliveira e Silva, Gaps between consecutive primes.
T. Oliveira e Silva, Tables of values of pi(x) and of pi2(x).
T. Oliveira e Silva, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060. - _Felix Fröhlich_, Jun 23 2014
Jörg Richstein, Verifying the Goldbach conjecture up to 4 * 10^14, Math. Comput., 70 (2001), 1745-1749.
Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Math. Comp. 61 (1993), pp. 931-934.
Eric Weisstein's World of Mathematics, Goldbach Conjecture.
A. Zaccagnini, Goldbach Variations: problems with prime numbers.
Index entries for sequences related to Goldbach conjecture

Essentially identical to A035026.
Cf. A002375 (unordered sums), A002374, A014092, A035026, A059998, A001031, A002373, A045917, A006307.
Cf. A065091, A010051.
Cf. A069360, A085090.

a002372 n = sum $ map (a010051 . (2*n -)) $ takeWhile (< 2*n) a065091_list
-- Reinhard Zumkeller, Oct 19 2011

A002372 := func; [A002372(n):n in[1..82]]; // Jason Kimberley, Sep 01 2011

a:=proc(n) local c,k; c:=0: for k from 1 to n do if isprime(2*k+1)=true and isprime(2*n-2*k-1)=true then c:=c+1 else c:=c fi od end: seq(a(n),n=1..82); # Emeric Deutsch, Jul 14 2004

For[lst={}; n=1, n<=100, n++, For[cnt=0; i=1, i<=2n-1, i++ If[OddQ[i]&&PrimeQ[i]&&PrimeQ[2n-i], cnt++ ]]; AppendTo[lst, cnt]]; lst
(* second program: *)
A002372[n_] := Module[{i = 0}, Do[If[PrimeQ[2 n - Prime@p], i++], {p, 2, PrimePi[2 n - 3]}]; i]; Array[A002372, 82] (* JungHwan Min, Aug 24 2016 *)
i[n_] := If[PrimeQ[2 n - 1], 2 n - 1, 0]; A085090 = Array[i, 82];
r[n_] := Table[A085090[[k]] + A085090[[n - k + 1]], {k, 1, n}];
countzeros[l_List] := Sum[KroneckerDelta[0, k], {k, l}];
Table[n - 2 countzeros[A085090[[1 ;; n]]] + countzeros[r[n]],
{n, 1, 82}] (* Fred Daniel Kline, Aug 13 2018 *)
countPrimes[n_] := Sum[KroneckerDelta[True, PrimeQ[2 m - 1],
PrimeQ[2 (n - m + 1) - 1]], {m, 1, n}]; Array[countPrimes, 82] (* Fred Daniel Kline, Oct 07 2018 *)

isop(n) = (n % 2) && isprime(n);
a(n) = n*=2; sum(i=1, n-1, isop(i)*isop(n-i)); \\ Michel Marcus, Aug 22 2014 and May 28 2020

from sympy import isprime, primerange
def a(n): return sum([1 for p in primerange(3, 2*n-2) if isprime(2*n-p)])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 23 2017

a(n) = A010051(n) + 2*A061357(n), n > 2. - R. J. Mathar, Aug 19 2013

More terms from Larry Reeves (larryr(AT)acm.org), Jun 13 2002

Edited by M. F. Hasler, May 03 2019

cp's OEIS Frontend

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

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A045919 Partial sum of Goldbach numbers A045917.

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A254713 All numbers k such that the number of distinct parts of all A045917(k) Goldbach partitions of 2k is prime.

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A343185 Numbers k such that 2*k is a multiple of A045917(k).

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A350455 T(n,k) is the k-th semiprime whose sum of prime factors equals 2n, triangle T(n,k), n>=2, 1<=k<=A045917(n), read by rows.

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A239282 a(n) = A045917(n)*prime(n).

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A245629 Numbers n such that A000203(2*n) divides 2*n*A045917(n).

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A002375 From Goldbach conjecture: number of decompositions of 2n into an unordered sum of two odd primes.

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A245629 Numbers n such that A000203(2n) divides 2n*A045917(n).