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A002375
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From Goldbach conjecture: number of decompositions of 2n into an unordered sum of two odd primes.
(Formerly M0104 N0040)
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41
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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; internal format)
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OFFSET
| 1,5
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COMMENTS
| 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.
a(n)=ceil((1/2)A002372(n)). - Emeric Deutsch, Jul 14 2004
The conjecture has been verified up to 3 x 10^17 (see MathWorld link). [From 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).
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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.
Matti K. Sinisalo, Math. Comp. 61 (1993), pp. 931-934.
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).
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LINKS
| H. J. Smith, Table of n, a(n) for n = 1..20000
J.-M. Deshouillers, H. J. J. te Riele and Y. Saouter, New Experimental Results Concerning the Goldbach Conjecture
M. Herkommer, Goldbach Conjecture Research
A. V. Kumchev and D. I. Tolev, An invitation to additive number theory
J. Richstein, Verifying Goldbach's Conjecture up to 4 x 10^14
V. Shevelev, Binary additive problems: recursions for numbers of representations [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jan 22 2009]
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
Alessandro Languasco, Alessandro Zaccagnini, The number of Goldbach representations of an integer, Nov 14, 2010.
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FORMULA
| From Halberstam and Richert : a(n)<(8+0(1))*c(n)*n/ln(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
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
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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.
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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
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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 *)
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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]];
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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.
Sequence in context: A053597 A094570 * A045917 A029379 A058776 A029228
Adjacent sequences: A002372 A002373 A002374 * A002376 A002377 A002378
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Beginning corrected by Paul Zimmermann Mar 15 1996
More terms from James A. Sellers (sellersj(AT)math.psu.edu)
Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Apr 20 2010
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