

A002372


Goldbach conjecture: number of decompositions of 2n into ordered sums of two odd primes.
(Formerly M0421 N0161)


52



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, 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, 11, 24, 7, 10, 20, 6, 14, 18, 11, 10, 16, 14, 15, 22, 11, 10, 24, 8, 16, 22, 9, 16, 20, 10
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OFFSET

1,4


COMMENTS

The weak form of this conjecture was proved by Helfgott (see link below).  T. D. Noe, May 14 2013
Goldbach conjectured in 1742 that for n >= 3, this sequence never vanishes. This is still unproved.
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
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 12201; it occurs after the prime 80873624627234849.
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 doublecheck 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.
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
For n > 2, a(n) = 2*A002375(n)  A010051(n).  Jason Kimberley, Aug 31 2011
a(n) = Sum_{p odd prime < 2*n} A010051(2*n  p).  Reinhard Zumkeller, Oct 19 2011
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


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 9.
R. K. Guy, Unsolved problems in number theory, second edition, SpringerVerlag, 1994.
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. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 79, 80.
N. Pipping, Neue Tafeln für das Goldbachsche Gesetz nebst Berichtigungen zu den Haussnerschen Tafeln, Finska VetenskapsSocieteten, Comment. Physico Math. 4 (No. 4, 1927), pp. 127.
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).
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 LA3106, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Sep 1964.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
Peter B. Borwein, Stephen K. K. Choi, Greg Martin, Charles L. Samuels, Polynomials whose reducibility is related to the Goldbach conjecture, arXiv:1408.4881 [math.NT], 2014 (see R(N) on page 1).
J. M. Deshouillers, H. J. J. te Riele, Y. Saouter, New experimental results concerning the Goldbach conjecture, preprint, Centrum Wiskunde & Informatica, 1998.
J. M. Deshouillers, H. J. J. te Riele, Y. Saouter, New experimental results concerning the Goldbach conjecture, Algorithmic number theory (Portland, OR, 1998), 204215, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998.
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 170, 1922.
H. A. Helfgott, Major arcs for Goldbach's theorem, arXiv:1305.2897 [math.NT], 20132014.
Yan Kun, Li Hou Biao, Divisor Goldbach Conjecture and its Partition Number, arXiv:1603.05233 [math.NT], 2016.
T. Oliveira e Silva, Goldbach conjecture verification
T. Oliveira e Silva, Gaps between consecutive primes
T. Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
T. Oliveira e Silva, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 20332060.  Felix Fröhlich, Jun 23 2014
Jörg Richstein, Verifying the Goldbach conjecture up to 4 * 10^14, Math. Comput., 70 (2001), 17451749.
Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Math. Comp. 61 (1993), pp. 931934.
Eric Weisstein's World of Mathematics, Goldbach Conjecture
A. Zaccagnini, Goldbach Variations: problems with prime numbers.
Index entries for sequences related to Goldbach conjecture


FORMULA

a(n) = A010051(n) + 2*A061357(n), n>2.  R. J. Mathar, Aug 19 2013


EXAMPLE

2 has no such decompositions, so a(1) = 0. Idem for 4, whence a(2) = 0.
6 = 3+3, so a(3)=1. 8 = 3+5 = 5+3, so a(4) = 2.
10 = 5+5 = 3+7 = 7+3, so a(5) = 3. 12 = 5+7 = 7+5; so a(6) = 2, etc.


MAPLE

a:=proc(n) local c, k; c:=0: for k from 1 to n do if isprime(2*k+1)=true and isprime(2*n2*k1)=true then c:=c+1 else c:=c fi od end: seq(a(n), n=1..82); # Emeric Deutsch, Jul 14 2004


MATHEMATICA

For[lst={}; n=1, n<=100, n++, For[cnt=0; i=1, i<=2n1, i++ If[OddQ[i]&&PrimeQ[i]&&PrimeQ[2ni], cnt++ ]]; AppendTo[lst, cnt]]; lst
(* second program: *)
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 *)
i[n_] := If[PrimeQ[2 n  1], 2 n  1, 0]; A085090 = Array[i, 82];
r[n_] := Table[A085090[[k]] + A085090[[n  k + 1]], {k, 1, n}];
countzeros[l_List] := Sum[KroneckerDelta[0, k], {k, l}];
Table[n  2 countzeros[A085090[[1 ;; n]]] + countzeros[r[n]],
{n, 1, 82}] (* Fred Daniel Kline, Aug 13 2018 *)
countPrimes[n_] := Sum[KroneckerDelta[True, PrimeQ[2 m  1],
PrimeQ[2 (n  m + 1)  1]], {m, 1, n}]; Array[countPrimes, 82] (* Fred Daniel Kline, Oct 07 2018 *)


PROG

(MAGMA) A002372 := func<n#[p:p in[3..2*n3]IsPrime(p)and IsPrime(2*np)]>; [A002372(n):n in[1..82]]; // Jason Kimberley, Sep 01 2011
(Haskell)
a002372 n = sum $ map (a010051 . (2*n )) $ takeWhile (< 2*n) a065091_list
 Reinhard Zumkeller, Oct 19 2011
(PARI) isop(n) = (n % 2) && isprime(n);
a(n) = n*=2; sum(i=1, n1, isop(i)*isop(ni)); \\ Michel Marcus, Aug 22 2014 and May 28 2020
(Python)
from sympy import isprime, primerange
def a(n): return sum([1 for p in primerange(3, 2*n2) if isprime(2*np)])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 23 2017


CROSSREFS

Essentially identical to A035026.
Cf. A002375 (unordered sums), A002374, A014092, A035026, A059998, A001031, A002373, A045917, A006307.
Cf. A065091, A010051.
Cf. A069360, A085090.
Sequence in context: A231070 A230252 A295424 * A035026 A224962 A173540
Adjacent sequences: A002369 A002370 A002371 * A002373 A002374 A002375


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jun 13 2002
Edited by M. F. Hasler, May 03 2019


STATUS

approved



