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A045917 From Goldbach problem: number of decompositions of 2n into unordered sums of two primes.

%I A045917 #164 Mar 10 2025 11:12:21
%S A045917 0,1,1,1,2,1,2,2,2,2,3,3,3,2,3,2,4,4,2,3,4,3,4,5,4,3,5,3,4,6,3,5,6,2,
%T A045917 5,6,5,5,7,4,5,8,5,4,9,4,5,7,3,6,8,5,6,8,6,7,10,6,6,12,4,5,10,3,7,9,6,
%U A045917 5,8,7,8,11,6,5,12,4,8,11,5,8,10,5,6,13,9,6,11,7,7,14,6,8,13,5,8,11,7,9
%N A045917 From Goldbach problem: number of decompositions of 2n into unordered sums of two primes.
%C A045917 Note that A002375 (which differs only at the n = 2 term) is the main entry for this sequence.
%C A045917 The graph of this sequence is called Goldbach's comet. - _David W. Wilson_, Mar 19 2012
%C A045917 This is the row length sequence of A182138, A184995 and A198292. - _Jason Kimberley_, Oct 03 2012
%C A045917 The Goldbach conjecture states that a(n) > 0 for n >= 2. - _Wolfdieter Lang_, May 14 2016
%C A045917 With the second Maple program, the command G(2n) yields all the unordered pairs of prime numbers having sum 2n; caveat: a pair {a,a} is listed as {a}. Example: G(26) yields {{13}, {3,23}, {7,19}}. The command G(100000) yields 810 pairs very fast. - _Emeric Deutsch_, Jan 03 2017
%C A045917 Conjecture: Let p denote any prime in any decomposition of 2n. 4 and 6 are the only numbers n such that 2n + p is prime for every p. - _Ivan N. Ianakiev_, Apr 06 2017
%C A045917 Conjecture: For all m >= 0, there exists at least one possible value of n such that a(n) = m. - _Ahmad J. Masad_, Jan 06 2018
%C A045917 The previous conjecture is related to the sequence A053033. - _Ahmad J. Masad_, Dec 09 2019
%C A045917 Conjecture: For each k >= 0, there exists a minimum sufficiently large number r that depends on k such that for each n >= r, a(n) > k. - _Ahmad J. Masad_, Jan 08 2020
%C A045917 Conjecture: If the previous conjecture is true, then for each m >= 0, the number of terms that are equal to (m+1) is larger than the number of terms that are equal to m. - _Ahmad J. Masad_, Jan 08 2020
%C A045917 Also, the number of equidistant prime pairs in Goldbach's Prime Triangle for integers n > 2. An equidistant prime pair is a pair of not necessarily different prime numbers (p1, p2) that have the same distance d >= 0 from an integer n, i.e., so that p1 = n - d and p2 = n + d. - _Jörg Winkelmann_, Mar 05 2025
%D A045917 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, MA, 1996, Chapter 12, pages 236-257.
%D A045917 H. Halberstam and H. E. Richert, 1974, "Sieve methods", Academic press, London, New York, San Francisco.
%H A045917 H. J. Smith, <a href="/A045917/b045917.txt">Table of n, a(n) for n = 1..20000</a>
%H A045917 M. Herkommer, <a href="http://www.petrospec-technologies.com/Herkommer/goldbach.htm">Goldbach Conjecture Research</a>.
%H A045917 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>.
%H A045917 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>.
%H A045917 G. Xiao, WIMS server, <a href="http://wims.unice.fr/~wims/en_tool~number~goldbach.en.html">Goldbach</a>
%H A045917 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A045917 J. Winkelmann, <a href="https://zenodo.org/records/14947270">Goldbach's Prime Triangle — A Recreational Math Journey with an Introduction to Equidistant Primes</a>.
%F A045917 From Halberstam and Richert: a(n) < (8+0(1))*c(n)*n/log(n)^2 where c(n) = Product_{p>2} (1 - 1/(p-1)^2)*Product_{p|n, p>2} (p-1)/(p-2). It is conjectured that the factor 8 can be replaced by 2. - _Benoit Cloitre_, May 16 2002
%F A045917 a(n) = ceiling(A035026(n) / 2) = (A035026(n) + A010051(n)) / 2.
%F A045917 a(n) = Sum_{i=2..n} floor(2/Omega(i*(2*n-i))). - _Wesley Ivan Hurt_, Jan 24 2013
%F A045917 a(n) = A224709(n) + (primepi(2n-2) - primepi(n-1)) + primepi(n) + 1 - n. - _Anthony Browne_, May 03 2016
%F A045917 a(n) = A224708(2n) - A224708(2n+1) + A010051(n). - _Anthony Browne_, Jun 26 2016
%F A045917 a(n) = Sum_{k=n*(n-1)/2+2..n*(n+1)/2} A064911(A105020(k-1)). - _Wesley Ivan Hurt_, Sep 11 2021
%F A045917 a(n) = omega(A362641(n)) = omega(A362640(n)). - _Wesley Ivan Hurt_, Apr 28 2023
%p A045917 A045917 := proc(n)
%p A045917     local a,i ;
%p A045917     a := 0 ;
%p A045917     for i from 1 to n do
%p A045917         if isprime(i) and isprime(2*n-i) then
%p A045917             a := a+1 ;
%p A045917         end if;
%p A045917     end do:
%p A045917     a ;
%p A045917 end proc: # _R. J. Mathar_, Jul 01 2013
%p A045917 # second Maple program:
%p A045917 G := proc (n) local g, j: g := {}: for j from 2 to (1/2)*n do if isprime(j) and isprime(n-j) then g := `union`(g, {{n-j, j}}) end if end do: g end proc: seq(nops(G(2*n)), n = 1 .. 98); # _Emeric Deutsch_, Jan 03 2017
%t A045917 f[n_] := Length[Select[2n - Prime[Range[PrimePi[n]]], PrimeQ]]; Table[ f[n], {n, 100}] (* Paul Abbott, Jan 11 2005 *)
%t A045917 nn = 10^2; ps = Boole[PrimeQ[Range[1,2*nn,2]]]; Join[{0,1}, Table[Sum[ps[[i]] ps[[n-i+1]], {i, Ceiling[n/2]}], {n, 3, nn}]] (* _T. D. Noe_, Apr 13 2011 *)
%o A045917 (PARI) a(n)=my(s);forprime(p=2,n,s+=isprime(2*n-p));s \\ _Charles R Greathouse IV_, Mar 27 2012
%o A045917 (Haskell)
%o A045917 a045917 n = sum $ map (a010051 . (2 * n -)) $ takeWhile (<= n) a000040_list
%o A045917 -- _Reinhard Zumkeller_, Sep 02 2013
%o A045917 (Python)
%o A045917 from sympy import isprime
%o A045917 def A045917(n):
%o A045917     x = 0
%o A045917     for i in range(2,n+1):
%o A045917         if isprime(i) and isprime(2*n-i):
%o A045917             x += 1
%o A045917     return x # _Chai Wah Wu_, Feb 24 2015
%o A045917 (Magma) [#RestrictedPartitions(2*n,2,Set(PrimesInInterval(1,2*n))):n in [1..100]]; // _Marius A. Burtea_, Jan 23 2020
%Y A045917 Cf. A002375 (the main entry for this sequence (which differs only at the n=2 term)).
%Y A045917 Cf. A023036 (first appearance of n), A000954 (last (assumed) appearance of n).
%Y A045917 Cf. also A000040, A182138, A184995, A185297, A187129, A010051, A198292, A035026, A224708, A224709.
%Y A045917 Cf. A064911, A105020.
%K A045917 easy,nice,nonn,look
%O A045917 1,5
%A A045917 _Felice Russo_