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%I M0421 N0161 #155 Sep 04 2023 16:37:47
%S 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,
%T 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,
%U 11,24,7,10,20,6,14,18,11,10,16,14,15,22,11,10,24,8,16,22,9,16,20,10
%N Goldbach conjecture: number of decompositions of 2n into ordered sums of two odd primes.
%C The weak form of this conjecture was proved by Helfgott (see link below). - _T. D. Noe_, May 14 2013
%C Goldbach conjectured in 1742 that for n >= 3, this sequence never vanishes. This is still unproved.
%C 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
%C 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.
%C 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.
%C 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
%C For n > 2, a(n) = 2*A002375(n) - A010051(n). - _Jason Kimberley_, Aug 31 2011
%C a(n) = Sum_{p odd prime < 2*n} A010051(2*n - p). - _Reinhard Zumkeller_, Oct 19 2011
%C 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
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 9.
%D R. K. Guy, Unsolved problems in number theory, second edition, Springer-Verlag, 1994.
%D 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 D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 79, 80.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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.
%H T. D. Noe, <a href="/A002372/b002372.txt">Table of n, a(n) for n = 1..10000</a>
%H Peter B. Borwein, Stephen K. K. Choi, Greg Martin, Charles L. Samuels, <a href="http://arxiv.org/abs/1408.4881">Polynomials whose reducibility is related to the Goldbach conjecture</a>, arXiv:1408.4881 [math.NT], 2014 (see R(N) on page 1).
%H J. -M. Deshouillers, H. J. J. te Riele, Y. Saouter, <a href="https://ir.cwi.nl/pub/1222">New experimental results concerning the Goldbach conjecture</a>, preprint, Centrum Wiskunde & Informatica, 1998.
%H J. -M. Deshouillers, H. J. J. te Riele, Y. Saouter, <a href="http://dx.doi.org/10.1007/BFb0054863">New experimental results concerning the Goldbach conjecture</a>, Algorithmic number theory (Portland, OR, 1998), 204-215, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998.
%H G. H. Hardy and J. E. Littlewood, <a href="http://dx.doi.org/10.1007/BF02403921">Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes</a>, Acta Mathematica, Vol. 44, pp. 1-70, 1922.
%H H. A. Helfgott, <a href="http://arxiv.org/abs/1305.2897">Major arcs for Goldbach's theorem</a>, arXiv:1305.2897 [math.NT], 2013-2014.
%H Yan Kun, Li Hou Biao, <a href="http://arxiv.org/abs/1603.05233">Divisor Goldbach Conjecture and its Partition Number</a>, arXiv:1603.05233 [math.NT], 2016.
%H T. Oliveira e Silva, <a href="http://sweet.ua.pt/tos/goldbach.html">Goldbach conjecture verification</a>.
%H T. Oliveira e Silva, <a href="http://sweet.ua.pt/tos/gaps.html">Gaps between consecutive primes</a>.
%H T. Oliveira e Silva, <a href="http://sweet.ua.pt/tos/primes.html">Tables of values of pi(x) and of pi2(x)</a>.
%H T. Oliveira e Silva, <a href="http://dx.doi.org/10.1090/S0025-5718-2013-02787-1">Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18</a>, Math. Comp., 83 (2014), 2033-2060. - _Felix Fröhlich_, Jun 23 2014
%H Jörg Richstein, <a href="http://dx.doi.org/10.1090/S0025-5718-00-01290-4">Verifying the Goldbach conjecture up to 4 * 10^14</a>, Math. Comput., 70 (2001), 1745-1749.
%H Matti K. Sinisalo, <a href="http://dx.doi.org/10.1090/S0025-5718-1993-1185250-6">Checking the Goldbach conjecture up to 4*10^11</a>, Math. Comp. 61 (1993), pp. 931-934.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachConjecture.html">Goldbach Conjecture</a>.
%H A. Zaccagnini, <a href="http://people.dmi.unipr.it/alessandro.zaccagnini/psfiles/papers/Goldbach_E.pdf">Goldbach Variations: problems with prime numbers</a>.
%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%F a(n) = A010051(n) + 2*A061357(n), n > 2. - _R. J. Mathar_, Aug 19 2013
%e 2 has no such decompositions, so a(1) = 0.
%e Idem for 4, whence a(2) = 0.
%e 6 = 3+3, so a(3) = 1.
%e 8 = 3+5 = 5+3, so a(4) = 2.
%e 10 = 5+5 = 3+7 = 7+3, so a(5) = 3.
%e 12 = 5+7 = 7+5; so a(6) = 2, etc.
%p 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
%t 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
%t (* second program: *)
%t 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 *)
%t i[n_] := If[PrimeQ[2 n - 1], 2 n - 1, 0]; A085090 = Array[i, 82];
%t r[n_] := Table[A085090[[k]] + A085090[[n - k + 1]], {k, 1, n}];
%t countzeros[l_List] := Sum[KroneckerDelta[0, k], {k, l}];
%t Table[n - 2 countzeros[A085090[[1 ;; n]]] + countzeros[r[n]],
%t {n, 1, 82}] (* _Fred Daniel Kline_, Aug 13 2018 *)
%t countPrimes[n_] := Sum[KroneckerDelta[True, PrimeQ[2 m - 1],
%t PrimeQ[2 (n - m + 1) - 1]], {m, 1, n}]; Array[countPrimes, 82] (* _Fred Daniel Kline_, Oct 07 2018 *)
%o (Magma) A002372 := func<n|#[p:p in[3..2*n-3]|IsPrime(p)and IsPrime(2*n-p)]>; [A002372(n):n in[1..82]]; // _Jason Kimberley_, Sep 01 2011
%o (Haskell)
%o a002372 n = sum $ map (a010051 . (2*n -)) $ takeWhile (< 2*n) a065091_list
%o -- _Reinhard Zumkeller_, Oct 19 2011
%o (PARI) isop(n) = (n % 2) && isprime(n);
%o a(n) = n*=2; sum(i=1, n-1, isop(i)*isop(n-i)); \\ _Michel Marcus_, Aug 22 2014 and May 28 2020
%o (Python)
%o from sympy import isprime, primerange
%o def a(n): return sum([1 for p in primerange(3, 2*n-2) if isprime(2*n-p)])
%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Apr 23 2017
%Y Essentially identical to A035026.
%Y Cf. A002375 (unordered sums), A002374, A014092, A035026, A059998, A001031, A002373, A045917, A006307.
%Y Cf. A065091, A010051.
%Y Cf. A069360, A085090.
%K nonn,nice,easy
%O 1,4
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 13 2002
%E Edited by _M. F. Hasler_, May 03 2019
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