A002375 - OEIS

A002375

From Goldbach conjecture: number of decompositions of 2n into an unordered sum of two odd primes.
(Formerly M0104 N0040)

172

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

COMMENTS

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

REFERENCES

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).

LINKS

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

FORMULA

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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 *)

PROG

(MuPAD) 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:

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

(PARI) 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

(Magma) A002375 := func<n|#[p:p in[3..n]|IsPrime(p)and IsPrime(2*n-p)]>; [A002375(n):n in[1..98]];

(Sage)

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

(Haskell)

a002375 n = sum $ map (a010051 . (2 * n -)) $ takeWhile (<= n) a065091_list