A155216 - OEIS

%I #32 Jul 21 2019 17:19:15

%S 0,1,2,2,2,3,3,4,3,4,5,5,6,7,4,6,6,7,8,8,7,8,9,8,8,10,9,10,10,10,13,

%T 11,10,12,11,12,12,14,12,13,14,13,13,15,13,15,15,17,16,15,15,15,16,18,

%U 16,16,18,17,19,17,20,19,19,18,18,20,19,20,21,20,18,22,21,22,20,23,19,22

%N Number of decompositions of positive even numbers 2n into unordered sums of a prime and a prime or semiprime (Chen's partitions).

%C According to Chen's result, the terms of this sequence are positive, at least for sufficiently large n.

%D J. R. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Kexue Tongbao, 17(1966), 385-386.

%D J. R. Chen, On the representation of a larger even integer as the sum of a prime and the product of at most two primes, Sci. Sinica, 16(1973), 157-176.

%D P. M. Ross, On Chen's theorem that each large even number has the form (p1+p2) or (p1+p2p3), J. London Math. Soc. (2) 10(1975), 500-506.

%H Peter J. C. Moses, <a href="/A155216/b155216.txt">Table of n, a(n) for n = 1..10000</a>

%H V. Shevelev, <a href="http://arxiv.org/abs/0901.3102">Binary additive problems: recursions for numbers of representations</a>

%F For n >= 2, a(n) = Sum_{3<=p<=n, p prime} A(2*n - p) + Sum_{t<=2*n, t odd semiprime} A(2*n - t) + A(n) - binomial(A(n),2) + delta(n) - a(n-1) - ... - a(1), where A(n) = A033270(n), delta(n) = 1, if n is prime, and delta(n) = 2, if n is a composite number. - _Vladimir Shevelev_, Jul 11 2013

%p A155216 := proc(n) local a,p,q,twon ; twon := 2*n ; a := 0 ; for i from 1 do p := ithprime(i) ; if ithprime(i) > twon then break; end if; q := twon -ithprime(i) ; if isprime(q) and q>= p then a := a+1 ; end if; end do: for i from 1 do p := ithprime(i) ; if ithprime(i) > twon then break; end if; q := twon -ithprime(i) ; if isA001358(q) then a := a+1 ; end if; end do: return a; end proc: seq(A155216(n),n=1..80) ; # _R. J. Mathar_, Jul 26 2010

%t a[n_] := Module[{k = 0, p, q}, For[i = 1, True, i++, p = Prime[i]; If[p > 2n, Break[]]; q = 2n - Prime[i]; If[PrimeQ[q] && q >= p, k++]]; For[i = 1, True, i++, p = Prime[i]; If[p > 2n, Break[]]; q = 2n - Prime[i]; If[ PrimeOmega[q] == 2, k++]]; k];

%t Array[a, 80] (* _Jean-François Alcover_, Nov 28 2017, after _R. J. Mathar_ *)

%Y Cf. A002375, A046927.

%K nonn

%O 1,3

%A _Vladimir Shevelev_, Jan 22 2009

%E Terms beyond a(21) from _R. J. Mathar_, Jul 26 2010