A096139 -id:A096139 - cp's OEIS frontend

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

N. J. A. Sloane

Number of decompositions of 2*n into sum of two noncomposite numbers. - Omar E. Pol, Dec 12 2024

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

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.

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

Cf. A002372 (the main entry), A002373, A002374, A002375, A006307, A008578, A010051, A045917.

a001031 n = sum (map a010051 gs) + fromEnum (1 `elem` gs)
   where gs = map (2 * n -) $ takeWhile (<= n) a008578_list
-- Reinhard Zumkeller, Aug 28 2013

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 *)

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

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

More terms from Ray Chandler, Sep 19 2003

A341982 Number of ways to write n as an ordered sum of 2 primes (counting 1 as a prime).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 4, 2, 3, 0, 4, 2, 5, 2, 4, 0, 6, 2, 6, 2, 5, 0, 8, 2, 5, 0, 4, 0, 8, 2, 6, 2, 7, 0, 8, 0, 5, 2, 6, 0, 10, 2, 8, 2, 7, 0, 12, 2, 8, 0, 6, 0, 12, 2, 6, 0, 7, 0, 14, 2, 7, 2, 10, 0, 12, 0, 6, 2, 10, 0, 14, 2, 11, 2, 10, 0, 14, 0, 10, 2, 9, 0, 18, 2, 9, 0, 8

Offset: 2

Ilya Gutkovskiy, Feb 24 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..10000

Cf. A008578, A073610, A096139, A280917, A283762, A341945, A341973, A341983, A341984.

b:= proc(n) option remember; series(`if`(n=0, 1, x*add(
      `if`(j=1 or isprime(j), b(n-j), 0), j=1..n)), x, 3)
    end:
a:= n-> coeff(b(n), x, 2):
seq(a(n), n=2..88);  # Alois P. Heinz, Feb 24 2021

nmax = 88; CoefficientList[Series[(x + Sum[x^Prime[k], {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &

G.f.: ( x + Sum_{k>=1} x^prime(k) )^2.

A347744 Number of compositions (ordered partitions) of n into at most 2 prime parts (counting 1 as a prime).

Original entry on oeis.org

1, 1, 2, 3, 3, 3, 3, 3, 4, 2, 3, 1, 4, 3, 5, 2, 4, 1, 6, 3, 6, 2, 5, 1, 8, 2, 5, 0, 4, 1, 8, 3, 6, 2, 7, 0, 8, 1, 5, 2, 6, 1, 10, 3, 8, 2, 7, 1, 12, 2, 8, 0, 6, 1, 12, 2, 6, 0, 7, 1, 14, 3, 7, 2, 10, 0, 12, 1, 6, 2, 10, 1, 14, 3, 11, 2, 10, 0, 14, 1, 10

Offset: 0

Ilya Gutkovskiy, Sep 11 2021

nonn

Cf. A001031, A008578, A096139, A280917, A341982, A347739, A347745.

Table[Length@Flatten[Permutations/@IntegerPartitions[n,2,Join[{1},Prime@Range@PrimePi@n]],1],{n,0,80}] (* Giorgos Kalogeropoulos, Sep 12 2021 *)

{ a(n) = local(s(n) = if(n<2,1,isprime(n))); if(n==0,1,sum(i=1,n,s(i)*s(n-i))); } \\ Christian Krause, Dec 06 2022

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A001031 Goldbach conjecture: a(n) = number of decompositions of 2n into sum of two primes (counting 1 as a prime).

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A341982 Number of ways to write n as an ordered sum of 2 primes (counting 1 as a prime).

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A347744 Number of compositions (ordered partitions) of n into at most 2 prime parts (counting 1 as a prime).

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