Search: id:A002375
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%I A002375 M0104 N0040
%S A002375 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,
%T A002375 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,
%U A002375 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
%N A002375 From Goldbach conjecture: number of different decompositions of 2n into 
               an unordered sum of two odd primes.
%C A002375 The Goldbach conjecture is that for n >= 3, this sequence is always positive.
%C A002375 This has been checked up to at least 10^18 - see A002372.
%C A002375 With the exception of the n=2 term, identical to A045917.
%C A002375 a(n)=ceil((1/2)A002372(n)). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Jul 14 2004
%C A002375 The conjecture has been verified up to 3 x 10^17 (see http://mathworld.wolfram.com/
               GoldbachConjecture.html). [From Dmitry Kamenetsky (dkamen(AT)rsise.anu.edu.au), 
               Oct 17 2008]
%D A002375 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers,
               " Perseus Books, Cambridge, MA, 1996, Chapter 12, Pages 236-257.
%D A002375 Apostolos K. Doxiadis, Uncle Petros and Goldbach's Conjecture, Bloomsbury 
               Pub. PLC USA, 2000.
%D A002375 D. A. Grave, Traktat z Algebrichnogo Analizu (Monograph on Algebraic 
               Analysis). Vol. 2, p. 19. Vidavnitstvo Akademiia Nauk, Kiev, 1938.
%D A002375 H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, 
               London, New York, San Francisco.
%D A002375 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 
               105, National Research Council, Washington, DC, 1941, p. 80.
%D A002375 Matti K. Sinisalo, Math. Comp. 61 (1993), pp. 931-934.
%D A002375 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002375 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002375 H. J. Smith, <a href="b002375.txt">Table of n, a(n) for n = 1..20000</
               a>
%H A002375 J.-M. Deshouillers, H. J. J. te Riele and Y. Saouter, <a href="http:/
               /db.cwi.nl/rapporten/abstract.php?abstractnr=768">New Experimental 
               Results Concerning the Goldbach Conjecture</a>
%H A002375 M. Herkommer, <a href="http://www.petrospec-technologies.com/Herkommer/
               goldbach.htm">Goldbach Conjecture Research</a>
%H A002375 A. V. Kumchev and D. I. Tolev, <a href="http://arXiv.org/abs/math.NT/
               0412220">An invitation to additive number theory</a>
%H A002375 J. Richstein, <a href="http://www.informatik.uni-giessen.de/staff/richstein/
               res/g-en.html">Verifying Goldbach's Conjecture up to 4 x 10^14</a>
%H A002375 V. Shevelev, <a href="http://arXiv.org/abs/math.NT/0901.3102">Binary 
               additive problems: recursions for numbers of representations</a> 
               [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Jan 22 2009]
%H A002375 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GoldbachPartition.html">Goldbach Partition</a>
%H A002375 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">
               Goldbach's conjecture</a>
%H A002375 G. Xiao, WIMS server, <a href="http://wims.unice.fr/~wims/en_tool~number~goldbach.en.html">
               Goldbach</a>
%H A002375 <a href="Sindx_Go.html#Goldbach">Index entries for sequences related 
               to Goldbach conjecture</a>
%F A002375 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 (benoit7848c(AT)orange.fr), May 20 
               2002
%F A002375 G.f.=Sum(Sum(x^(p(i)+p(j)), i=2..j), j=2..infinity), where p(k) is the 
               k-th prime. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 27 2007
%e A002375 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.
%p A002375 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)];
%p A002375 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)
%p A002375 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 (deutsch(AT)duke.poly.edu), Aug 
               27 2007
%t A002375 f[n_] := Length[ Select[2n - Prime[ Range[2, PrimePi[n]]], PrimeQ]]; 
               Table[ f[n], {n, 100}] (Paul Abbott, Jan 11 2005)
%o A002375 (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:
%o A002375 (PARI) A002375(n)=sum(i=2,primepi(n),isprime(2*n-prime(i))) /* ...i=1... 
               gives A045917 */
%o A002375 (PARI) for(n=1,100,print1(sum(i=2,n,sum(j=2,i,if(prime(i)+prime(j)-2*n,
               0,1))),","))
%Y A002375 See also A061358. Cf. A002372 (ordered sums), A002373, A002374, A045917.
%Y A002375 A023036 is (essentially) the first appearance of n and A000954 is the 
               last (assumed) appearance of n.
%Y A002375 Sequence in context: A129363 A053597 A094570 this_sequence A045917 A029379 
               A058776
%Y A002375 Adjacent sequences: A002372 A002373 A002374 this_sequence A002376 A002377 
               A002378
%K A002375 nonn,easy,nice
%O A002375 1,5
%A A002375 N. J. A. Sloane (njas(AT)research.att.com).
%E A002375 Beginning corrected by Paul Zimmermann Mar 15 1996
%E A002375 More terms from James A. Sellers (sellersj(AT)math.psu.edu)

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