A001031 Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).
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
Offset: 1
Examples
1 is counted as a prime, so a(1)=1 since 2=1+1, a(2)=2 since 4=2+2=3+1, ..
References
- T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 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), 204-215, 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, Springer-Verlag, 1994.
- 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 LA-3106, 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
- 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 Math., Vol. 44, No. 1 (1923), pp. 1-70.
- 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. 1745-1749, 2001.
- Matti K. Sinisalo, Checking the Goldbach conjecture up to 4*10^11, Mathematics of Computation, Vol. 61, No. 204, pp. 931-934, October 1993.
- Eric Weisstein's World of Mathematics, Goldbach Partition
- Index entries for sequences related to Goldbach conjecture
Crossrefs
Programs
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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
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Mathematica
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 *) -
PARI
a(n)=my(s); forprime(p=2,n, if(isprime(2*n-p), s++)); if(isprime(2*n-1), s+1, s) \\ Charles R Greathouse IV, Feb 06 2017
Formula
Not very efficient: a(n) = (Sum_{i=1..n} (pi(i) - pi(i-1))*(pi(2*n-i) - pi(2*n-i-1))) + (pi(2*n-1) - pi(2*n-2)) + floor(1/n). - Wesley Ivan Hurt, Jan 06 2013
a(n) = floor((A096139(n)+1)/2). - Reinhard Zumkeller, Aug 28 2013
Extensions
More terms from Ray Chandler, Sep 19 2003
Comments