login
This site is supported by donations to The OEIS Foundation.

 

Logo

110 people attended OEIS-50 (videos, suggestions); annual fundraising drive to start soon (donate); editors, please edit! (stack is over 300), your editing is more valuable than any donation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002375 From Goldbach conjecture: number of decompositions of 2n into an unordered sum of two odd primes.
(Formerly M0104 N0040)
113
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 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).

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. 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

T. D. Noe and 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.

H. A. Helfgott, Minor arcs for Goldbach's problem, arXiv:1205.5252

H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897

H. A. Helfgott, The ternary Goldbach conjecture is true, arxiv:1312.7748, 2013.

H. A. Helfgott, The ternary Goldbach problem, arXiv:1404.2224, 2014.

M. Herkommer, Goldbach Conjecture Research

A. V. Kumchev and D. I. Tolev, An invitation to additive number theory, arXiv:math.NT/0412220

Alessandro Languasco, Alessandro Zaccagnini, The number of Goldbach representations of an integer, arXiv:1011.3198.

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:math.NT/0901.3102

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) = prod(p > 2, (1-1/(p-1)^2))*prod(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) = ceil(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( ((pi(i) - pi(i-1)) * (pi(2n-i) - pi(2n-i-1)) ), i = 1..n) - (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)]; # Deutsch

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 *)

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) for(n=1, 100, print1(sum(i=2, n, sum(j=2, i, if(prime(i)+prime(j)-2*n, 0, 1))), ", "))

(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 = map(lambda p: 2*n - p, P)

    F = filter(is_prime, M)

    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

-- Reinhard Zumkeller, Sep 02 2013

CROSSREFS

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).

Sequence in context: A225638 A230443 * A045917 A240708 A235645 A240874

Adjacent sequences:  A002372 A002373 A002374 * A002376 A002377 A002378

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Apr 30 1991

EXTENSIONS

Beginning corrected by Paul Zimmermann, Mar 15 1996

More terms from James A. Sellers

Edited by Charles R Greathouse IV, Apr 20 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 25 23:29 EDT 2014. Contains 248566 sequences.