

A001031


Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).
(Formerly M0213 N0077)


20



1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 3, 4, 3, 2, 4, 3, 4, 4, 3, 3, 5, 4, 4, 6, 4, 3, 6, 3, 4, 7, 4, 5, 6, 3, 5, 7, 6, 5, 7, 5, 5, 9, 5, 4, 10, 4, 5, 7, 4, 6, 9, 6, 6, 9, 7, 7, 11, 6, 6, 12, 4, 5, 10, 4, 7, 10, 6, 5, 9, 8, 8, 11, 6, 5, 13, 5, 8, 11, 6, 8, 10, 6, 6, 14, 9, 6, 12, 7, 7, 15, 7, 8, 13, 5, 8, 12, 8, 9
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

a(n) = floor((A096139(n)+1)/2).  Reinhard Zumkeller, Aug 28 2013


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 9.
Deshouillers, J.M.; te Riele, H. J. J.; and Saouter, Y.; New experimental results concerning the Goldbach conjecture. Algorithmic number theory (Portland, OR, 1998), 204215, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998.
Apostolos Doxiadis: Uncle Petros and Goldbach's Conjecture, Faber and Faber, 2001
R. K. Guy, Unsolved problems in number theory, second edition, SpringerVerlag, 1994.
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.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 79.
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
Romeo Meštrović, Different classes of binary necklaces and a combinatorial method for their enumerations, arXiv:1804.00992 [math.CO], 2018.
T. Oliveira e Silva, Goldbach conjecture verification
J. Richstein, Verifying the Goldbach conjecture up to 4*10^14, Mathematics of Computation, Vol. 70, No. 236, pp. 17451749, 2001.
Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Mathematics of Computation, Vol. 61, No. 204, pp. 931934, October 1993.
Eric Weisstein's World of Mathematics, Goldbach Partition
Index entries for sequences related to Goldbach conjecture


FORMULA

Not very efficient: a(n) = (Sum_{i=1..n} (pi(i)  pi(i1))*(pi(2*ni)  pi(2*ni1))) + (pi(2*n1)  pi(2*n2)) + floor(1/n).  Wesley Ivan Hurt, Jan 06 2013


EXAMPLE

1 is counted as a prime, so a(1)=1 since 2=1+1, a(2)=2 since 4=2+2=3+1, ..


MATHEMATICA

a[n_] := Length @ Select[PowersRepresentations[2 n, 2, 1], (#[[1]] == 1  PrimeQ[#[[1]]]) && (#[[2]] == 1  PrimeQ[#[[2]]]) &]; Array[a, 98] (* JeanFrançois Alcover, Apr 11 2011 *)
nn = 10^2; ps = Boole[PrimeQ[Range[2*nn]]]; ps[[1]] = 1; Table[Sum[ps[[i]] ps[[2*n  i]], {i, n}], {n, nn}] (* T. D. Noe, Apr 11 2011 *)


PROG

(Haskell)
a001031 n = sum (map a010051 gs) + fromEnum (1 `elem` gs)
where gs = map (2 * n ) $ takeWhile (<= n) a008578_list
 Reinhard Zumkeller, Aug 28 2013
(PARI) a(n)=my(s); forprime(p=2, n, if(isprime(2*np), s++)); if(isprime(2*n1), s+1, s) \\ Charles R Greathouse IV, Feb 06 2017


CROSSREFS

Cf. A002372 (the main entry), A002373, A002374, A002375, A006307, A008578, A010051, A045917.
Sequence in context: A189172 A286888 A257212 * A035250 A165054 A067743
Adjacent sequences: A001028 A001029 A001030 * A001032 A001033 A001034


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Ray Chandler, Sep 19 2003


STATUS

approved



